This project implements different predictive modelling procedures for dichotomous categorical responses using various helpful packages in Python. Models applied in the analysis to predict dichotomous categorical responses included the Logistic Regression, Decision Trees, Random Forest, Naive Bayes and Support Vector Machine algorithms. Remedial procedures on addressing class imbalance including Class Weighting, Synthetic Minority Oversampling Technique and Condensed Nearest Neighbors were similarly considered, as applicable. Ensemble learning using Stacking which consolidate many different models types on the same data and using another model to learn how to best combine the predictions was also explored. All results were consolidated in a Summary presented at the end of the document.
Binary classification learning refers to a predictive modelling problem where only two class labels are predicted for a given sample of input data. These models use the training data set and calculate how to best map instances of input data to the specific class labels. Typically, binary classification tasks involve one class that is the normal state (assigned the class label 0) and another class that is the abnormal state (assigned the class label 1). It is common to structure a binary classification task with a model that predicts a Bernoulli probability distribution for each instance. The Bernoulli distribution is a discrete probability distribution that covers a case where an event will have a binary outcome as either a 0 or 1. For a binary classification, this means that the model predicts a probability of an instance belonging to class 1, or the abnormal state. The algorithms applied in this study attempt to categorize the input data and form dichotomous groups based on their similarities.
Datasets used for the analysis were separately gathered and consolidated from various sources including:
This study hypothesized that various global development indicators and indices influence cancer rates across countries.
The target variable for the study is:
The predictor variables for the study are:
##################################
# Loading Python Libraries
##################################
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
import itertools
%matplotlib inline
from operator import add,mul,truediv
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PowerTransformer
from sklearn.preprocessing import StandardScaler
from scipy import stats
from sklearn.linear_model import LogisticRegression
from sklearn.tree import DecisionTreeClassifier
from sklearn.ensemble import RandomForestClassifier
from sklearn.svm import SVC
from sklearn.metrics import accuracy_score,precision_score,recall_score,f1_score,roc_auc_score
from sklearn.model_selection import train_test_split, GridSearchCV
##################################
# Loading the dataset
##################################
cancer_rate = pd.read_csv('CategoricalCancerRates.csv')
##################################
# Performing a general exploration of the dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate.shape)
Dataset Dimensions:
(177, 22)
##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate.dtypes)
Column Names and Data Types:
COUNTRY object CANRAT object GDPPER float64 URBPOP float64 PATRES float64 RNDGDP float64 POPGRO float64 LIFEXP float64 TUBINC float64 DTHCMD float64 AGRLND float64 GHGEMI float64 RELOUT float64 METEMI float64 FORARE float64 CO2EMI float64 PM2EXP float64 POPDEN float64 ENRTER float64 GDPCAP float64 HDICAT object EPISCO float64 dtype: object
##################################
# Taking a snapshot of the dataset
##################################
cancer_rate.head()
| COUNTRY | CANRAT | GDPPER | URBPOP | PATRES | RNDGDP | POPGRO | LIFEXP | TUBINC | DTHCMD | ... | RELOUT | METEMI | FORARE | CO2EMI | PM2EXP | POPDEN | ENRTER | GDPCAP | HDICAT | EPISCO | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | Australia | High | 98380.63601 | 86.241 | 2368.0 | NaN | 1.235701 | 83.200000 | 7.2 | 4.941054 | ... | 13.637841 | 131484.763200 | 17.421315 | 14.772658 | 24.893584 | 3.335312 | 110.139221 | 51722.06900 | VH | 60.1 |
| 1 | New Zealand | High | 77541.76438 | 86.699 | 348.0 | NaN | 2.204789 | 82.256098 | 7.2 | 4.354730 | ... | 80.081439 | 32241.937000 | 37.570126 | 6.160799 | NaN | 19.331586 | 75.734833 | 41760.59478 | VH | 56.7 |
| 2 | Ireland | High | 198405.87500 | 63.653 | 75.0 | 1.23244 | 1.029111 | 82.556098 | 5.3 | 5.684596 | ... | 27.965408 | 15252.824630 | 11.351720 | 6.768228 | 0.274092 | 72.367281 | 74.680313 | 85420.19086 | VH | 57.4 |
| 3 | United States | High | 130941.63690 | 82.664 | 269586.0 | 3.42287 | 0.964348 | 76.980488 | 2.3 | 5.302060 | ... | 13.228593 | 748241.402900 | 33.866926 | 13.032828 | 3.343170 | 36.240985 | 87.567657 | 63528.63430 | VH | 51.1 |
| 4 | Denmark | High | 113300.60110 | 88.116 | 1261.0 | 2.96873 | 0.291641 | 81.602439 | 4.1 | 6.826140 | ... | 65.505925 | 7778.773921 | 15.711000 | 4.691237 | 56.914456 | 145.785100 | 82.664330 | 60915.42440 | VH | 77.9 |
5 rows × 22 columns
##################################
# Setting the levels of the categorical variables
##################################
cancer_rate['CANRAT'] = cancer_rate['CANRAT'].astype('category')
cancer_rate['CANRAT'] = cancer_rate['CANRAT'].cat.set_categories(['Low', 'High'], ordered=True)
cancer_rate['HDICAT'] = cancer_rate['HDICAT'].astype('category')
cancer_rate['HDICAT'] = cancer_rate['HDICAT'].cat.set_categories(['L', 'M', 'H', 'VH'], ordered=True)
##################################
# Performing a general exploration of the numeric variables
##################################
print('Numeric Variable Summary:')
display(cancer_rate.describe(include='number').transpose())
Numeric Variable Summary:
| count | mean | std | min | 25% | 50% | 75% | max | |
|---|---|---|---|---|---|---|---|---|
| GDPPER | 165.0 | 45284.424283 | 3.941794e+04 | 1718.804896 | 13545.254510 | 34024.900890 | 66778.416050 | 2.346469e+05 |
| URBPOP | 174.0 | 59.788121 | 2.280640e+01 | 13.345000 | 42.432750 | 61.701500 | 79.186500 | 1.000000e+02 |
| PATRES | 108.0 | 20607.388889 | 1.340683e+05 | 1.000000 | 35.250000 | 244.500000 | 1297.750000 | 1.344817e+06 |
| RNDGDP | 74.0 | 1.197474 | 1.189956e+00 | 0.039770 | 0.256372 | 0.873660 | 1.608842 | 5.354510e+00 |
| POPGRO | 174.0 | 1.127028 | 1.197718e+00 | -2.079337 | 0.236900 | 1.179959 | 2.031154 | 3.727101e+00 |
| LIFEXP | 174.0 | 71.746113 | 7.606209e+00 | 52.777000 | 65.907500 | 72.464610 | 77.523500 | 8.456000e+01 |
| TUBINC | 174.0 | 105.005862 | 1.367229e+02 | 0.770000 | 12.000000 | 44.500000 | 147.750000 | 5.920000e+02 |
| DTHCMD | 170.0 | 21.260521 | 1.927333e+01 | 1.283611 | 6.078009 | 12.456279 | 36.980457 | 6.520789e+01 |
| AGRLND | 174.0 | 38.793456 | 2.171551e+01 | 0.512821 | 20.130276 | 40.386649 | 54.013754 | 8.084112e+01 |
| GHGEMI | 170.0 | 259582.709895 | 1.118550e+06 | 179.725150 | 12527.487367 | 41009.275980 | 116482.578575 | 1.294287e+07 |
| RELOUT | 153.0 | 39.760036 | 3.191492e+01 | 0.000296 | 10.582691 | 32.381668 | 63.011450 | 1.000000e+02 |
| METEMI | 170.0 | 47876.133575 | 1.346611e+05 | 11.596147 | 3662.884908 | 11118.976025 | 32368.909040 | 1.186285e+06 |
| FORARE | 173.0 | 32.218177 | 2.312001e+01 | 0.008078 | 11.604388 | 31.509048 | 49.071780 | 9.741212e+01 |
| CO2EMI | 170.0 | 3.751097 | 4.606479e+00 | 0.032585 | 0.631924 | 2.298368 | 4.823496 | 3.172684e+01 |
| PM2EXP | 167.0 | 91.940595 | 2.206003e+01 | 0.274092 | 99.627134 | 100.000000 | 100.000000 | 1.000000e+02 |
| POPDEN | 174.0 | 200.886765 | 6.453834e+02 | 2.115134 | 27.454539 | 77.983133 | 153.993650 | 7.918951e+03 |
| ENRTER | 116.0 | 49.994997 | 2.970619e+01 | 2.432581 | 22.107195 | 53.392460 | 71.057467 | 1.433107e+02 |
| GDPCAP | 170.0 | 13992.095610 | 1.957954e+04 | 216.827417 | 1870.503029 | 5348.192875 | 17421.116227 | 1.173705e+05 |
| EPISCO | 165.0 | 42.946667 | 1.249086e+01 | 18.900000 | 33.000000 | 40.900000 | 50.500000 | 7.790000e+01 |
##################################
# Performing a general exploration of the object variable
##################################
print('Object Variable Summary:')
display(cancer_rate.describe(include='object').transpose())
Object Variable Summary:
| count | unique | top | freq | |
|---|---|---|---|---|
| COUNTRY | 177 | 177 | Australia | 1 |
##################################
# Performing a general exploration of the categorical variables
##################################
print('Categorical Variable Summary:')
display(cancer_rate.describe(include='category').transpose())
Categorical Variable Summary:
| count | unique | top | freq | |
|---|---|---|---|---|
| CANRAT | 177 | 2 | Low | 132 |
| HDICAT | 167 | 4 | VH | 59 |
##################################
# Performing a general exploration of the response variable
##################################
cancer_rate.CANRAT.value_counts(normalize = True)
Low 0.745763 High 0.254237 Name: CANRAT, dtype: float64
Data quality findings based on assessment are as follows:
##################################
# Counting the number of duplicated rows
##################################
cancer_rate.duplicated().sum()
0
##################################
# Gathering the data types for each column
##################################
data_type_list = list(cancer_rate.dtypes)
##################################
# Gathering the variable names for each column
##################################
variable_name_list = list(cancer_rate.columns)
##################################
# Gathering the number of observations for each column
##################################
row_count_list = list([len(cancer_rate)] * len(cancer_rate.columns))
##################################
# Gathering the number of missing data for each column
##################################
null_count_list = list(cancer_rate.isna().sum(axis=0))
##################################
# Gathering the number of non-missing data for each column
##################################
non_null_count_list = list(cancer_rate.count())
##################################
# Gathering the missing data percentage for each column
##################################
fill_rate_list = map(truediv, non_null_count_list, row_count_list)
##################################
# Formulating the summary
# for all columns
##################################
all_column_quality_summary = pd.DataFrame(zip(variable_name_list,
data_type_list,
row_count_list,
non_null_count_list,
null_count_list,
fill_rate_list),
columns=['Column.Name',
'Column.Type',
'Row.Count',
'Non.Null.Count',
'Null.Count',
'Fill.Rate'])
display(all_column_quality_summary)
| Column.Name | Column.Type | Row.Count | Non.Null.Count | Null.Count | Fill.Rate | |
|---|---|---|---|---|---|---|
| 0 | COUNTRY | object | 177 | 177 | 0 | 1.000000 |
| 1 | CANRAT | category | 177 | 177 | 0 | 1.000000 |
| 2 | GDPPER | float64 | 177 | 165 | 12 | 0.932203 |
| 3 | URBPOP | float64 | 177 | 174 | 3 | 0.983051 |
| 4 | PATRES | float64 | 177 | 108 | 69 | 0.610169 |
| 5 | RNDGDP | float64 | 177 | 74 | 103 | 0.418079 |
| 6 | POPGRO | float64 | 177 | 174 | 3 | 0.983051 |
| 7 | LIFEXP | float64 | 177 | 174 | 3 | 0.983051 |
| 8 | TUBINC | float64 | 177 | 174 | 3 | 0.983051 |
| 9 | DTHCMD | float64 | 177 | 170 | 7 | 0.960452 |
| 10 | AGRLND | float64 | 177 | 174 | 3 | 0.983051 |
| 11 | GHGEMI | float64 | 177 | 170 | 7 | 0.960452 |
| 12 | RELOUT | float64 | 177 | 153 | 24 | 0.864407 |
| 13 | METEMI | float64 | 177 | 170 | 7 | 0.960452 |
| 14 | FORARE | float64 | 177 | 173 | 4 | 0.977401 |
| 15 | CO2EMI | float64 | 177 | 170 | 7 | 0.960452 |
| 16 | PM2EXP | float64 | 177 | 167 | 10 | 0.943503 |
| 17 | POPDEN | float64 | 177 | 174 | 3 | 0.983051 |
| 18 | ENRTER | float64 | 177 | 116 | 61 | 0.655367 |
| 19 | GDPCAP | float64 | 177 | 170 | 7 | 0.960452 |
| 20 | HDICAT | category | 177 | 167 | 10 | 0.943503 |
| 21 | EPISCO | float64 | 177 | 165 | 12 | 0.932203 |
##################################
# Counting the number of columns
# with Fill.Rate < 1.00
##################################
len(all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<1)])
20
##################################
# Identifying the columns
# with Fill.Rate < 1.00
##################################
display(all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<1)].sort_values(by=['Fill.Rate'], ascending=True))
| Column.Name | Column.Type | Row.Count | Non.Null.Count | Null.Count | Fill.Rate | |
|---|---|---|---|---|---|---|
| 5 | RNDGDP | float64 | 177 | 74 | 103 | 0.418079 |
| 4 | PATRES | float64 | 177 | 108 | 69 | 0.610169 |
| 18 | ENRTER | float64 | 177 | 116 | 61 | 0.655367 |
| 12 | RELOUT | float64 | 177 | 153 | 24 | 0.864407 |
| 2 | GDPPER | float64 | 177 | 165 | 12 | 0.932203 |
| 21 | EPISCO | float64 | 177 | 165 | 12 | 0.932203 |
| 20 | HDICAT | category | 177 | 167 | 10 | 0.943503 |
| 16 | PM2EXP | float64 | 177 | 167 | 10 | 0.943503 |
| 9 | DTHCMD | float64 | 177 | 170 | 7 | 0.960452 |
| 13 | METEMI | float64 | 177 | 170 | 7 | 0.960452 |
| 15 | CO2EMI | float64 | 177 | 170 | 7 | 0.960452 |
| 19 | GDPCAP | float64 | 177 | 170 | 7 | 0.960452 |
| 11 | GHGEMI | float64 | 177 | 170 | 7 | 0.960452 |
| 14 | FORARE | float64 | 177 | 173 | 4 | 0.977401 |
| 8 | TUBINC | float64 | 177 | 174 | 3 | 0.983051 |
| 10 | AGRLND | float64 | 177 | 174 | 3 | 0.983051 |
| 6 | POPGRO | float64 | 177 | 174 | 3 | 0.983051 |
| 17 | POPDEN | float64 | 177 | 174 | 3 | 0.983051 |
| 3 | URBPOP | float64 | 177 | 174 | 3 | 0.983051 |
| 7 | LIFEXP | float64 | 177 | 174 | 3 | 0.983051 |
##################################
# Identifying the rows
# with Fill.Rate < 0.90
##################################
column_low_fill_rate = all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<0.90)]
##################################
# Gathering the metadata labels for each observation
##################################
row_metadata_list = cancer_rate["COUNTRY"].values.tolist()
##################################
# Gathering the number of columns for each observation
##################################
column_count_list = list([len(cancer_rate.columns)] * len(cancer_rate))
##################################
# Gathering the number of missing data for each row
##################################
null_row_list = list(cancer_rate.isna().sum(axis=1))
##################################
# Gathering the missing data percentage for each column
##################################
missing_rate_list = map(truediv, null_row_list, column_count_list)
##################################
# Identifying the rows
# with missing data
##################################
all_row_quality_summary = pd.DataFrame(zip(row_metadata_list,
column_count_list,
null_row_list,
missing_rate_list),
columns=['Row.Name',
'Column.Count',
'Null.Count',
'Missing.Rate'])
display(all_row_quality_summary)
| Row.Name | Column.Count | Null.Count | Missing.Rate | |
|---|---|---|---|---|
| 0 | Australia | 22 | 1 | 0.045455 |
| 1 | New Zealand | 22 | 2 | 0.090909 |
| 2 | Ireland | 22 | 0 | 0.000000 |
| 3 | United States | 22 | 0 | 0.000000 |
| 4 | Denmark | 22 | 0 | 0.000000 |
| ... | ... | ... | ... | ... |
| 172 | Congo Republic | 22 | 3 | 0.136364 |
| 173 | Bhutan | 22 | 2 | 0.090909 |
| 174 | Nepal | 22 | 2 | 0.090909 |
| 175 | Gambia | 22 | 4 | 0.181818 |
| 176 | Niger | 22 | 2 | 0.090909 |
177 rows × 4 columns
##################################
# Counting the number of rows
# with Missing.Rate > 0.00
##################################
len(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.00)])
120
##################################
# Counting the number of rows
# with Missing.Rate > 0.20
##################################
len(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)])
14
##################################
# Identifying the rows
# with Missing.Rate > 0.20
##################################
row_high_missing_rate = all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)]
##################################
# Identifying the rows
# with Missing.Rate > 0.20
##################################
display(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)].sort_values(by=['Missing.Rate'], ascending=False))
| Row.Name | Column.Count | Null.Count | Missing.Rate | |
|---|---|---|---|---|
| 35 | Guadeloupe | 22 | 20 | 0.909091 |
| 39 | Martinique | 22 | 20 | 0.909091 |
| 56 | French Guiana | 22 | 20 | 0.909091 |
| 13 | New Caledonia | 22 | 11 | 0.500000 |
| 44 | French Polynesia | 22 | 11 | 0.500000 |
| 75 | Guam | 22 | 11 | 0.500000 |
| 53 | Puerto Rico | 22 | 9 | 0.409091 |
| 85 | North Korea | 22 | 6 | 0.272727 |
| 132 | Somalia | 22 | 6 | 0.272727 |
| 168 | South Sudan | 22 | 6 | 0.272727 |
| 73 | Venezuela | 22 | 5 | 0.227273 |
| 117 | Libya | 22 | 5 | 0.227273 |
| 161 | Eritrea | 22 | 5 | 0.227273 |
| 164 | Yemen | 22 | 5 | 0.227273 |
##################################
# Formulating the dataset
# with numeric columns only
##################################
cancer_rate_numeric = cancer_rate.select_dtypes(include='number')
##################################
# Gathering the variable names for each numeric column
##################################
numeric_variable_name_list = cancer_rate_numeric.columns
##################################
# Gathering the minimum value for each numeric column
##################################
numeric_minimum_list = cancer_rate_numeric.min()
##################################
# Gathering the mean value for each numeric column
##################################
numeric_mean_list = cancer_rate_numeric.mean()
##################################
# Gathering the median value for each numeric column
##################################
numeric_median_list = cancer_rate_numeric.median()
##################################
# Gathering the maximum value for each numeric column
##################################
numeric_maximum_list = cancer_rate_numeric.max()
##################################
# Gathering the first mode values for each numeric column
##################################
numeric_first_mode_list = [cancer_rate[x].value_counts(dropna=True).index.tolist()[0] for x in cancer_rate_numeric]
##################################
# Gathering the second mode values for each numeric column
##################################
numeric_second_mode_list = [cancer_rate[x].value_counts(dropna=True).index.tolist()[1] for x in cancer_rate_numeric]
##################################
# Gathering the count of first mode values for each numeric column
##################################
numeric_first_mode_count_list = [cancer_rate_numeric[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_numeric]
##################################
# Gathering the count of second mode values for each numeric column
##################################
numeric_second_mode_count_list = [cancer_rate_numeric[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_numeric]
##################################
# Gathering the first mode to second mode ratio for each numeric column
##################################
numeric_first_second_mode_ratio_list = map(truediv, numeric_first_mode_count_list, numeric_second_mode_count_list)
##################################
# Gathering the count of unique values for each numeric column
##################################
numeric_unique_count_list = cancer_rate_numeric.nunique(dropna=True)
##################################
# Gathering the number of observations for each numeric column
##################################
numeric_row_count_list = list([len(cancer_rate_numeric)] * len(cancer_rate_numeric.columns))
##################################
# Gathering the unique to count ratio for each numeric column
##################################
numeric_unique_count_ratio_list = map(truediv, numeric_unique_count_list, numeric_row_count_list)
##################################
# Gathering the skewness value for each numeric column
##################################
numeric_skewness_list = cancer_rate_numeric.skew()
##################################
# Gathering the kurtosis value for each numeric column
##################################
numeric_kurtosis_list = cancer_rate_numeric.kurtosis()
numeric_column_quality_summary = pd.DataFrame(zip(numeric_variable_name_list,
numeric_minimum_list,
numeric_mean_list,
numeric_median_list,
numeric_maximum_list,
numeric_first_mode_list,
numeric_second_mode_list,
numeric_first_mode_count_list,
numeric_second_mode_count_list,
numeric_first_second_mode_ratio_list,
numeric_unique_count_list,
numeric_row_count_list,
numeric_unique_count_ratio_list,
numeric_skewness_list,
numeric_kurtosis_list),
columns=['Numeric.Column.Name',
'Minimum',
'Mean',
'Median',
'Maximum',
'First.Mode',
'Second.Mode',
'First.Mode.Count',
'Second.Mode.Count',
'First.Second.Mode.Ratio',
'Unique.Count',
'Row.Count',
'Unique.Count.Ratio',
'Skewness',
'Kurtosis'])
display(numeric_column_quality_summary)
| Numeric.Column.Name | Minimum | Mean | Median | Maximum | First.Mode | Second.Mode | First.Mode.Count | Second.Mode.Count | First.Second.Mode.Ratio | Unique.Count | Row.Count | Unique.Count.Ratio | Skewness | Kurtosis | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | GDPPER | 1718.804896 | 45284.424283 | 34024.900890 | 2.346469e+05 | 98380.636010 | 42154.178100 | 1 | 1 | 1.000000 | 165 | 177 | 0.932203 | 1.517574 | 3.471992 |
| 1 | URBPOP | 13.345000 | 59.788121 | 61.701500 | 1.000000e+02 | 100.000000 | 52.516000 | 2 | 1 | 2.000000 | 173 | 177 | 0.977401 | -0.210702 | -0.962847 |
| 2 | PATRES | 1.000000 | 20607.388889 | 244.500000 | 1.344817e+06 | 6.000000 | 2.000000 | 4 | 3 | 1.333333 | 97 | 177 | 0.548023 | 9.284436 | 91.187178 |
| 3 | RNDGDP | 0.039770 | 1.197474 | 0.873660 | 5.354510e+00 | 1.232440 | 0.962180 | 1 | 1 | 1.000000 | 74 | 177 | 0.418079 | 1.396742 | 1.695957 |
| 4 | POPGRO | -2.079337 | 1.127028 | 1.179959 | 3.727101e+00 | 1.235701 | 1.483129 | 1 | 1 | 1.000000 | 174 | 177 | 0.983051 | -0.195161 | -0.423580 |
| 5 | LIFEXP | 52.777000 | 71.746113 | 72.464610 | 8.456000e+01 | 83.200000 | 68.687000 | 1 | 1 | 1.000000 | 174 | 177 | 0.983051 | -0.357965 | -0.649601 |
| 6 | TUBINC | 0.770000 | 105.005862 | 44.500000 | 5.920000e+02 | 12.000000 | 7.200000 | 4 | 3 | 1.333333 | 131 | 177 | 0.740113 | 1.746333 | 2.429368 |
| 7 | DTHCMD | 1.283611 | 21.260521 | 12.456279 | 6.520789e+01 | 4.941054 | 42.079403 | 1 | 1 | 1.000000 | 170 | 177 | 0.960452 | 0.900509 | -0.691541 |
| 8 | AGRLND | 0.512821 | 38.793456 | 40.386649 | 8.084112e+01 | 46.252480 | 72.006469 | 1 | 1 | 1.000000 | 174 | 177 | 0.983051 | 0.074000 | -0.926249 |
| 9 | GHGEMI | 179.725150 | 259582.709895 | 41009.275980 | 1.294287e+07 | 571903.119900 | 3000.932259 | 1 | 1 | 1.000000 | 170 | 177 | 0.960452 | 9.496120 | 101.637308 |
| 10 | RELOUT | 0.000296 | 39.760036 | 32.381668 | 1.000000e+02 | 100.000000 | 13.637841 | 3 | 1 | 3.000000 | 151 | 177 | 0.853107 | 0.501088 | -0.981774 |
| 11 | METEMI | 11.596147 | 47876.133575 | 11118.976025 | 1.186285e+06 | 131484.763200 | 1326.034028 | 1 | 1 | 1.000000 | 170 | 177 | 0.960452 | 5.801014 | 38.661386 |
| 12 | FORARE | 0.008078 | 32.218177 | 31.509048 | 9.741212e+01 | 17.421315 | 8.782159 | 1 | 1 | 1.000000 | 173 | 177 | 0.977401 | 0.519277 | -0.322589 |
| 13 | CO2EMI | 0.032585 | 3.751097 | 2.298368 | 3.172684e+01 | 14.772658 | 0.972088 | 1 | 1 | 1.000000 | 170 | 177 | 0.960452 | 2.721552 | 10.311574 |
| 14 | PM2EXP | 0.274092 | 91.940595 | 100.000000 | 1.000000e+02 | 100.000000 | 100.000000 | 106 | 2 | 53.000000 | 61 | 177 | 0.344633 | -3.141557 | 9.032386 |
| 15 | POPDEN | 2.115134 | 200.886765 | 77.983133 | 7.918951e+03 | 3.335312 | 13.300785 | 1 | 1 | 1.000000 | 174 | 177 | 0.983051 | 10.267750 | 119.995256 |
| 16 | ENRTER | 2.432581 | 49.994997 | 53.392460 | 1.433107e+02 | 110.139221 | 45.220661 | 1 | 1 | 1.000000 | 116 | 177 | 0.655367 | 0.275863 | -0.392895 |
| 17 | GDPCAP | 216.827417 | 13992.095610 | 5348.192875 | 1.173705e+05 | 51722.069000 | 3961.726633 | 1 | 1 | 1.000000 | 170 | 177 | 0.960452 | 2.258568 | 5.938690 |
| 18 | EPISCO | 18.900000 | 42.946667 | 40.900000 | 7.790000e+01 | 29.600000 | 43.600000 | 3 | 3 | 1.000000 | 137 | 177 | 0.774011 | 0.641799 | 0.035208 |
##################################
# Counting the number of numeric columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['First.Second.Mode.Ratio']>5)])
1
##################################
# Identifying the numeric columns
# with First.Second.Mode.Ratio > 5.00
##################################
display(numeric_column_quality_summary[(numeric_column_quality_summary['First.Second.Mode.Ratio']>5)].sort_values(by=['First.Second.Mode.Ratio'], ascending=False))
| Numeric.Column.Name | Minimum | Mean | Median | Maximum | First.Mode | Second.Mode | First.Mode.Count | Second.Mode.Count | First.Second.Mode.Ratio | Unique.Count | Row.Count | Unique.Count.Ratio | Skewness | Kurtosis | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 14 | PM2EXP | 0.274092 | 91.940595 | 100.0 | 100.0 | 100.0 | 100.0 | 106 | 2 | 53.0 | 61 | 177 | 0.344633 | -3.141557 | 9.032386 |
##################################
# Counting the number of numeric columns
# with Unique.Count.Ratio > 10.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['Unique.Count.Ratio']>10)])
0
##################################
# Counting the number of numeric columns
# with Skewness > 3.00 or Skewness < -3.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['Skewness']>3) | (numeric_column_quality_summary['Skewness']<(-3))])
5
##################################
# Identifying the numeric columns
# with Skewness > 3.00 or Skewness < -3.00
##################################
display(numeric_column_quality_summary[(numeric_column_quality_summary['Skewness']>3) | (numeric_column_quality_summary['Skewness']<(-3))].sort_values(by=['Skewness'], ascending=False))
| Numeric.Column.Name | Minimum | Mean | Median | Maximum | First.Mode | Second.Mode | First.Mode.Count | Second.Mode.Count | First.Second.Mode.Ratio | Unique.Count | Row.Count | Unique.Count.Ratio | Skewness | Kurtosis | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 15 | POPDEN | 2.115134 | 200.886765 | 77.983133 | 7.918951e+03 | 3.335312 | 13.300785 | 1 | 1 | 1.000000 | 174 | 177 | 0.983051 | 10.267750 | 119.995256 |
| 9 | GHGEMI | 179.725150 | 259582.709895 | 41009.275980 | 1.294287e+07 | 571903.119900 | 3000.932259 | 1 | 1 | 1.000000 | 170 | 177 | 0.960452 | 9.496120 | 101.637308 |
| 2 | PATRES | 1.000000 | 20607.388889 | 244.500000 | 1.344817e+06 | 6.000000 | 2.000000 | 4 | 3 | 1.333333 | 97 | 177 | 0.548023 | 9.284436 | 91.187178 |
| 11 | METEMI | 11.596147 | 47876.133575 | 11118.976025 | 1.186285e+06 | 131484.763200 | 1326.034028 | 1 | 1 | 1.000000 | 170 | 177 | 0.960452 | 5.801014 | 38.661386 |
| 14 | PM2EXP | 0.274092 | 91.940595 | 100.000000 | 1.000000e+02 | 100.000000 | 100.000000 | 106 | 2 | 53.000000 | 61 | 177 | 0.344633 | -3.141557 | 9.032386 |
##################################
# Formulating the dataset
# with object column only
##################################
cancer_rate_object = cancer_rate.select_dtypes(include='object')
##################################
# Gathering the variable names for the object column
##################################
object_variable_name_list = cancer_rate_object.columns
##################################
# Gathering the first mode values for the object column
##################################
object_first_mode_list = [cancer_rate[x].value_counts().index.tolist()[0] for x in cancer_rate_object]
##################################
# Gathering the second mode values for each object column
##################################
object_second_mode_list = [cancer_rate[x].value_counts().index.tolist()[1] for x in cancer_rate_object]
##################################
# Gathering the count of first mode values for each object column
##################################
object_first_mode_count_list = [cancer_rate_object[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_object]
##################################
# Gathering the count of second mode values for each object column
##################################
object_second_mode_count_list = [cancer_rate_object[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_object]
##################################
# Gathering the first mode to second mode ratio for each object column
##################################
object_first_second_mode_ratio_list = map(truediv, object_first_mode_count_list, object_second_mode_count_list)
##################################
# Gathering the count of unique values for each object column
##################################
object_unique_count_list = cancer_rate_object.nunique(dropna=True)
##################################
# Gathering the number of observations for each object column
##################################
object_row_count_list = list([len(cancer_rate_object)] * len(cancer_rate_object.columns))
##################################
# Gathering the unique to count ratio for each object column
##################################
object_unique_count_ratio_list = map(truediv, object_unique_count_list, object_row_count_list)
object_column_quality_summary = pd.DataFrame(zip(object_variable_name_list,
object_first_mode_list,
object_second_mode_list,
object_first_mode_count_list,
object_second_mode_count_list,
object_first_second_mode_ratio_list,
object_unique_count_list,
object_row_count_list,
object_unique_count_ratio_list),
columns=['Object.Column.Name',
'First.Mode',
'Second.Mode',
'First.Mode.Count',
'Second.Mode.Count',
'First.Second.Mode.Ratio',
'Unique.Count',
'Row.Count',
'Unique.Count.Ratio'])
display(object_column_quality_summary)
| Object.Column.Name | First.Mode | Second.Mode | First.Mode.Count | Second.Mode.Count | First.Second.Mode.Ratio | Unique.Count | Row.Count | Unique.Count.Ratio | |
|---|---|---|---|---|---|---|---|---|---|
| 0 | COUNTRY | Australia | Mauritius | 1 | 1 | 1.0 | 177 | 177 | 1.0 |
##################################
# Counting the number of object columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(object_column_quality_summary[(object_column_quality_summary['First.Second.Mode.Ratio']>5)])
0
##################################
# Counting the number of object columns
# with Unique.Count.Ratio > 10.00
##################################
len(object_column_quality_summary[(object_column_quality_summary['Unique.Count.Ratio']>10)])
0
##################################
# Formulating the dataset
# with categorical columns only
##################################
cancer_rate_categorical = cancer_rate.select_dtypes(include='category')
##################################
# Gathering the variable names for the categorical column
##################################
categorical_variable_name_list = cancer_rate_categorical.columns
##################################
# Gathering the first mode values for each categorical column
##################################
categorical_first_mode_list = [cancer_rate[x].value_counts().index.tolist()[0] for x in cancer_rate_categorical]
##################################
# Gathering the second mode values for each categorical column
##################################
categorical_second_mode_list = [cancer_rate[x].value_counts().index.tolist()[1] for x in cancer_rate_categorical]
##################################
# Gathering the count of first mode values for each categorical column
##################################
categorical_first_mode_count_list = [cancer_rate_categorical[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_categorical]
##################################
# Gathering the count of second mode values for each categorical column
##################################
categorical_second_mode_count_list = [cancer_rate_categorical[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_categorical]
##################################
# Gathering the first mode to second mode ratio for each categorical column
##################################
categorical_first_second_mode_ratio_list = map(truediv, categorical_first_mode_count_list, categorical_second_mode_count_list)
##################################
# Gathering the count of unique values for each categorical column
##################################
categorical_unique_count_list = cancer_rate_categorical.nunique(dropna=True)
##################################
# Gathering the number of observations for each categorical column
##################################
categorical_row_count_list = list([len(cancer_rate_categorical)] * len(cancer_rate_categorical.columns))
##################################
# Gathering the unique to count ratio for each categorical column
##################################
categorical_unique_count_ratio_list = map(truediv, categorical_unique_count_list, categorical_row_count_list)
categorical_column_quality_summary = pd.DataFrame(zip(categorical_variable_name_list,
categorical_first_mode_list,
categorical_second_mode_list,
categorical_first_mode_count_list,
categorical_second_mode_count_list,
categorical_first_second_mode_ratio_list,
categorical_unique_count_list,
categorical_row_count_list,
categorical_unique_count_ratio_list),
columns=['Categorical.Column.Name',
'First.Mode',
'Second.Mode',
'First.Mode.Count',
'Second.Mode.Count',
'First.Second.Mode.Ratio',
'Unique.Count',
'Row.Count',
'Unique.Count.Ratio'])
display(categorical_column_quality_summary)
| Categorical.Column.Name | First.Mode | Second.Mode | First.Mode.Count | Second.Mode.Count | First.Second.Mode.Ratio | Unique.Count | Row.Count | Unique.Count.Ratio | |
|---|---|---|---|---|---|---|---|---|---|
| 0 | CANRAT | Low | High | 132 | 45 | 2.933333 | 2 | 177 | 0.011299 |
| 1 | HDICAT | VH | H | 59 | 39 | 1.512821 | 4 | 177 | 0.022599 |
##################################
# Counting the number of categorical columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(categorical_column_quality_summary[(categorical_column_quality_summary['First.Second.Mode.Ratio']>5)])
0
##################################
# Counting the number of categorical columns
# with Unique.Count.Ratio > 10.00
##################################
len(categorical_column_quality_summary[(categorical_column_quality_summary['Unique.Count.Ratio']>10)])
0
##################################
# Performing a general exploration of the original dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate.shape)
Dataset Dimensions:
(177, 22)
##################################
# Filtering out the rows with
# with Missing.Rate > 0.20
##################################
cancer_rate_filtered_row = cancer_rate.drop(cancer_rate[cancer_rate.COUNTRY.isin(row_high_missing_rate['Row.Name'].values.tolist())].index)
##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_filtered_row.shape)
Dataset Dimensions:
(163, 22)
##################################
# Filtering out the columns with
# with Fill.Rate < 0.90
##################################
cancer_rate_filtered_row_column = cancer_rate_filtered_row.drop(column_low_fill_rate['Column.Name'].values.tolist(), axis=1)
##################################
# Formulating a new dataset object
# for the cleaned data
##################################
cancer_rate_cleaned = cancer_rate_filtered_row_column
##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_cleaned.shape)
Dataset Dimensions:
(163, 18)
Iterative Imputer is based on the Multivariate Imputation by Chained Equations (MICE) algorithm - an imputation method based on fully conditional specification, where each incomplete variable is imputed by a separate model. As a sequential regression imputation technique, the algorithm imputes an incomplete column (target column) by generating plausible synthetic values given other columns in the data. Each incomplete column must act as a target column, and has its own specific set of predictors. For predictors that are incomplete themselves, the most recently generated imputations are used to complete the predictors prior to prior to imputation of the target columns.
Linear Regression explores the linear relationship between a scalar response and one or more covariates by having the conditional mean of the dependent variable be an affine function of the independent variables. The relationship is modeled through a disturbance term which represents an unobserved random variable that adds noise. The algorithm is typically formulated from the data using the least squares method which seeks to estimate the coefficients by minimizing the squared residual function. The linear equation assigns one scale factor represented by a coefficient to each covariate and an additional coefficient called the intercept or the bias coefficient which gives the line an additional degree of freedom allowing to move up and down a two-dimensional plot.
##################################
# Formulating the summary
# for all cleaned columns
##################################
cleaned_column_quality_summary = pd.DataFrame(zip(list(cancer_rate_cleaned.columns),
list(cancer_rate_cleaned.dtypes),
list([len(cancer_rate_cleaned)] * len(cancer_rate_cleaned.columns)),
list(cancer_rate_cleaned.count()),
list(cancer_rate_cleaned.isna().sum(axis=0))),
columns=['Column.Name',
'Column.Type',
'Row.Count',
'Non.Null.Count',
'Null.Count'])
display(cleaned_column_quality_summary)
| Column.Name | Column.Type | Row.Count | Non.Null.Count | Null.Count | |
|---|---|---|---|---|---|
| 0 | COUNTRY | object | 163 | 163 | 0 |
| 1 | CANRAT | category | 163 | 163 | 0 |
| 2 | GDPPER | float64 | 163 | 162 | 1 |
| 3 | URBPOP | float64 | 163 | 163 | 0 |
| 4 | POPGRO | float64 | 163 | 163 | 0 |
| 5 | LIFEXP | float64 | 163 | 163 | 0 |
| 6 | TUBINC | float64 | 163 | 163 | 0 |
| 7 | DTHCMD | float64 | 163 | 163 | 0 |
| 8 | AGRLND | float64 | 163 | 163 | 0 |
| 9 | GHGEMI | float64 | 163 | 163 | 0 |
| 10 | METEMI | float64 | 163 | 163 | 0 |
| 11 | FORARE | float64 | 163 | 162 | 1 |
| 12 | CO2EMI | float64 | 163 | 163 | 0 |
| 13 | PM2EXP | float64 | 163 | 158 | 5 |
| 14 | POPDEN | float64 | 163 | 163 | 0 |
| 15 | GDPCAP | float64 | 163 | 163 | 0 |
| 16 | HDICAT | category | 163 | 162 | 1 |
| 17 | EPISCO | float64 | 163 | 163 | 0 |
##################################
# Formulating the cleaned dataset
# with categorical columns only
##################################
cancer_rate_cleaned_categorical = cancer_rate_cleaned.select_dtypes(include='object')
##################################
# Formulating the cleaned dataset
# with numeric columns only
##################################
cancer_rate_cleaned_numeric = cancer_rate_cleaned.select_dtypes(include='number')
##################################
# Taking a snapshot of the cleaned dataset
##################################
cancer_rate_cleaned_numeric.head()
| GDPPER | URBPOP | POPGRO | LIFEXP | TUBINC | DTHCMD | AGRLND | GHGEMI | METEMI | FORARE | CO2EMI | PM2EXP | POPDEN | GDPCAP | EPISCO | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 98380.63601 | 86.241 | 1.235701 | 83.200000 | 7.2 | 4.941054 | 46.252480 | 5.719031e+05 | 131484.763200 | 17.421315 | 14.772658 | 24.893584 | 3.335312 | 51722.06900 | 60.1 |
| 1 | 77541.76438 | 86.699 | 2.204789 | 82.256098 | 7.2 | 4.354730 | 38.562911 | 8.015803e+04 | 32241.937000 | 37.570126 | 6.160799 | NaN | 19.331586 | 41760.59478 | 56.7 |
| 2 | 198405.87500 | 63.653 | 1.029111 | 82.556098 | 5.3 | 5.684596 | 65.495718 | 5.949773e+04 | 15252.824630 | 11.351720 | 6.768228 | 0.274092 | 72.367281 | 85420.19086 | 57.4 |
| 3 | 130941.63690 | 82.664 | 0.964348 | 76.980488 | 2.3 | 5.302060 | 44.363367 | 5.505181e+06 | 748241.402900 | 33.866926 | 13.032828 | 3.343170 | 36.240985 | 63528.63430 | 51.1 |
| 4 | 113300.60110 | 88.116 | 0.291641 | 81.602439 | 4.1 | 6.826140 | 65.499675 | 4.113555e+04 | 7778.773921 | 15.711000 | 4.691237 | 56.914456 | 145.785100 | 60915.42440 | 77.9 |
##################################
# Defining the estimator to be used
# at each step of the round-robin imputation
##################################
lr = LinearRegression()
##################################
# Defining the parameter of the
# iterative imputer which will estimate
# the columns with missing values
# as a function of the other columns
# in a round-robin fashion
##################################
iterative_imputer = IterativeImputer(
estimator = lr,
max_iter = 10,
tol = 1e-10,
imputation_order = 'ascending',
random_state=88888888
)
##################################
# Implementing the iterative imputer
##################################
cancer_rate_imputed_numeric_array = iterative_imputer.fit_transform(cancer_rate_cleaned_numeric)
##################################
# Transforming the imputed data
# from an array to a dataframe
##################################
cancer_rate_imputed_numeric = pd.DataFrame(cancer_rate_imputed_numeric_array,
columns = cancer_rate_cleaned_numeric.columns)
##################################
# Taking a snapshot of the imputed dataset
##################################
cancer_rate_imputed_numeric.head()
| GDPPER | URBPOP | POPGRO | LIFEXP | TUBINC | DTHCMD | AGRLND | GHGEMI | METEMI | FORARE | CO2EMI | PM2EXP | POPDEN | GDPCAP | EPISCO | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 98380.63601 | 86.241 | 1.235701 | 83.200000 | 7.2 | 4.941054 | 46.252480 | 5.719031e+05 | 131484.763200 | 17.421315 | 14.772658 | 24.893584 | 3.335312 | 51722.06900 | 60.1 |
| 1 | 77541.76438 | 86.699 | 2.204789 | 82.256098 | 7.2 | 4.354730 | 38.562911 | 8.015803e+04 | 32241.937000 | 37.570126 | 6.160799 | 65.867296 | 19.331586 | 41760.59478 | 56.7 |
| 2 | 198405.87500 | 63.653 | 1.029111 | 82.556098 | 5.3 | 5.684596 | 65.495718 | 5.949773e+04 | 15252.824630 | 11.351720 | 6.768228 | 0.274092 | 72.367281 | 85420.19086 | 57.4 |
| 3 | 130941.63690 | 82.664 | 0.964348 | 76.980488 | 2.3 | 5.302060 | 44.363367 | 5.505181e+06 | 748241.402900 | 33.866926 | 13.032828 | 3.343170 | 36.240985 | 63528.63430 | 51.1 |
| 4 | 113300.60110 | 88.116 | 0.291641 | 81.602439 | 4.1 | 6.826140 | 65.499675 | 4.113555e+04 | 7778.773921 | 15.711000 | 4.691237 | 56.914456 | 145.785100 | 60915.42440 | 77.9 |
##################################
# Formulating the cleaned dataset
# with categorical columns only
##################################
cancer_rate_cleaned_categorical = cancer_rate_cleaned.select_dtypes(include='category')
##################################
# Imputing the missing data
# for categorical columns with
# the most frequent category
##################################
cancer_rate_cleaned_categorical['HDICAT'].fillna(cancer_rate_cleaned_categorical['HDICAT'].mode()[0], inplace=True)
cancer_rate_imputed_categorical = cancer_rate_cleaned_categorical.reset_index(drop=True)
##################################
# Formulating the imputed dataset
##################################
cancer_rate_imputed = pd.concat([cancer_rate_imputed_numeric,cancer_rate_imputed_categorical], axis=1, join='inner')
##################################
# Gathering the data types for each column
##################################
data_type_list = list(cancer_rate_imputed.dtypes)
##################################
# Gathering the variable names for each column
##################################
variable_name_list = list(cancer_rate_imputed.columns)
##################################
# Gathering the number of observations for each column
##################################
row_count_list = list([len(cancer_rate_imputed)] * len(cancer_rate_imputed.columns))
##################################
# Gathering the number of missing data for each column
##################################
null_count_list = list(cancer_rate_imputed.isna().sum(axis=0))
##################################
# Gathering the number of non-missing data for each column
##################################
non_null_count_list = list(cancer_rate_imputed.count())
##################################
# Gathering the missing data percentage for each column
##################################
fill_rate_list = map(truediv, non_null_count_list, row_count_list)
##################################
# Formulating the summary
# for all imputed columns
##################################
imputed_column_quality_summary = pd.DataFrame(zip(variable_name_list,
data_type_list,
row_count_list,
non_null_count_list,
null_count_list,
fill_rate_list),
columns=['Column.Name',
'Column.Type',
'Row.Count',
'Non.Null.Count',
'Null.Count',
'Fill.Rate'])
display(imputed_column_quality_summary)
| Column.Name | Column.Type | Row.Count | Non.Null.Count | Null.Count | Fill.Rate | |
|---|---|---|---|---|---|---|
| 0 | GDPPER | float64 | 163 | 163 | 0 | 1.0 |
| 1 | URBPOP | float64 | 163 | 163 | 0 | 1.0 |
| 2 | POPGRO | float64 | 163 | 163 | 0 | 1.0 |
| 3 | LIFEXP | float64 | 163 | 163 | 0 | 1.0 |
| 4 | TUBINC | float64 | 163 | 163 | 0 | 1.0 |
| 5 | DTHCMD | float64 | 163 | 163 | 0 | 1.0 |
| 6 | AGRLND | float64 | 163 | 163 | 0 | 1.0 |
| 7 | GHGEMI | float64 | 163 | 163 | 0 | 1.0 |
| 8 | METEMI | float64 | 163 | 163 | 0 | 1.0 |
| 9 | FORARE | float64 | 163 | 163 | 0 | 1.0 |
| 10 | CO2EMI | float64 | 163 | 163 | 0 | 1.0 |
| 11 | PM2EXP | float64 | 163 | 163 | 0 | 1.0 |
| 12 | POPDEN | float64 | 163 | 163 | 0 | 1.0 |
| 13 | GDPCAP | float64 | 163 | 163 | 0 | 1.0 |
| 14 | EPISCO | float64 | 163 | 163 | 0 | 1.0 |
| 15 | CANRAT | category | 163 | 163 | 0 | 1.0 |
| 16 | HDICAT | category | 163 | 163 | 0 | 1.0 |
##################################
# Formulating the imputed dataset
# with numeric columns only
##################################
cancer_rate_imputed_numeric = cancer_rate_imputed.select_dtypes(include='number')
##################################
# Gathering the variable names for each numeric column
##################################
numeric_variable_name_list = list(cancer_rate_imputed_numeric.columns)
##################################
# Gathering the skewness value for each numeric column
##################################
numeric_skewness_list = cancer_rate_imputed_numeric.skew()
##################################
# Computing the interquartile range
# for all columns
##################################
cancer_rate_imputed_numeric_q1 = cancer_rate_imputed_numeric.quantile(0.25)
cancer_rate_imputed_numeric_q3 = cancer_rate_imputed_numeric.quantile(0.75)
cancer_rate_imputed_numeric_iqr = cancer_rate_imputed_numeric_q3 - cancer_rate_imputed_numeric_q1
##################################
# Gathering the outlier count for each numeric column
# based on the interquartile range criterion
##################################
numeric_outlier_count_list = ((cancer_rate_imputed_numeric < (cancer_rate_imputed_numeric_q1 - 1.5 * cancer_rate_imputed_numeric_iqr)) | (cancer_rate_imputed_numeric > (cancer_rate_imputed_numeric_q3 + 1.5 * cancer_rate_imputed_numeric_iqr))).sum()
##################################
# Gathering the number of observations for each column
##################################
numeric_row_count_list = list([len(cancer_rate_imputed_numeric)] * len(cancer_rate_imputed_numeric.columns))
##################################
# Gathering the unique to count ratio for each categorical column
##################################
numeric_outlier_ratio_list = map(truediv, numeric_outlier_count_list, numeric_row_count_list)
##################################
# Formulating the outlier summary
# for all numeric columns
##################################
numeric_column_outlier_summary = pd.DataFrame(zip(numeric_variable_name_list,
numeric_skewness_list,
numeric_outlier_count_list,
numeric_row_count_list,
numeric_outlier_ratio_list),
columns=['Numeric.Column.Name',
'Skewness',
'Outlier.Count',
'Row.Count',
'Outlier.Ratio'])
display(numeric_column_outlier_summary)
| Numeric.Column.Name | Skewness | Outlier.Count | Row.Count | Outlier.Ratio | |
|---|---|---|---|---|---|
| 0 | GDPPER | 1.554457 | 3 | 163 | 0.018405 |
| 1 | URBPOP | -0.212327 | 0 | 163 | 0.000000 |
| 2 | POPGRO | -0.181666 | 0 | 163 | 0.000000 |
| 3 | LIFEXP | -0.329704 | 0 | 163 | 0.000000 |
| 4 | TUBINC | 1.747962 | 12 | 163 | 0.073620 |
| 5 | DTHCMD | 0.930709 | 0 | 163 | 0.000000 |
| 6 | AGRLND | 0.035315 | 0 | 163 | 0.000000 |
| 7 | GHGEMI | 9.299960 | 27 | 163 | 0.165644 |
| 8 | METEMI | 5.688689 | 20 | 163 | 0.122699 |
| 9 | FORARE | 0.563015 | 0 | 163 | 0.000000 |
| 10 | CO2EMI | 2.693585 | 11 | 163 | 0.067485 |
| 11 | PM2EXP | -3.088403 | 37 | 163 | 0.226994 |
| 12 | POPDEN | 9.972806 | 20 | 163 | 0.122699 |
| 13 | GDPCAP | 2.311079 | 22 | 163 | 0.134969 |
| 14 | EPISCO | 0.635994 | 3 | 163 | 0.018405 |
##################################
# Formulating the individual boxplots
# for all numeric columns
##################################
for column in cancer_rate_imputed_numeric:
plt.figure(figsize=(17,1))
sns.boxplot(data=cancer_rate_imputed_numeric, x=column)
Pearson’s Correlation Coefficient is a parametric measure of the linear correlation for a pair of features by calculating the ratio between their covariance and the product of their standard deviations. The presence of high absolute correlation values indicate the univariate association between the numeric predictors and the numeric response.
##################################
# Formulating a function
# to plot the correlation matrix
# for all pairwise combinations
# of numeric columns
##################################
def plot_correlation_matrix(corr, mask=None):
f, ax = plt.subplots(figsize=(11, 9))
sns.heatmap(corr,
ax=ax,
mask=mask,
annot=True,
vmin=-1,
vmax=1,
center=0,
cmap='coolwarm',
linewidths=1,
linecolor='gray',
cbar_kws={'orientation': 'horizontal'})
##################################
# Computing the correlation coefficients
# and correlation p-values
# among pairs of numeric columns
##################################
cancer_rate_imputed_numeric_correlation_pairs = {}
cancer_rate_imputed_numeric_columns = cancer_rate_imputed_numeric.columns.tolist()
for numeric_column_a, numeric_column_b in itertools.combinations(cancer_rate_imputed_numeric_columns, 2):
cancer_rate_imputed_numeric_correlation_pairs[numeric_column_a + '_' + numeric_column_b] = stats.pearsonr(
cancer_rate_imputed_numeric.loc[:, numeric_column_a],
cancer_rate_imputed_numeric.loc[:, numeric_column_b])
##################################
# Formulating the pairwise correlation summary
# for all numeric columns
##################################
cancer_rate_imputed_numeric_summary = cancer_rate_imputed_numeric.from_dict(cancer_rate_imputed_numeric_correlation_pairs, orient='index')
cancer_rate_imputed_numeric_summary.columns = ['Pearson.Correlation.Coefficient', 'Correlation.PValue']
display(cancer_rate_imputed_numeric_summary.sort_values(by=['Pearson.Correlation.Coefficient'], ascending=False).head(20))
| Pearson.Correlation.Coefficient | Correlation.PValue | |
|---|---|---|
| GDPPER_GDPCAP | 0.921010 | 8.158179e-68 |
| GHGEMI_METEMI | 0.905121 | 1.087643e-61 |
| POPGRO_DTHCMD | 0.759470 | 7.124695e-32 |
| GDPPER_LIFEXP | 0.755787 | 2.055178e-31 |
| GDPCAP_EPISCO | 0.696707 | 5.312642e-25 |
| LIFEXP_GDPCAP | 0.683834 | 8.321371e-24 |
| GDPPER_EPISCO | 0.680812 | 1.555304e-23 |
| GDPPER_URBPOP | 0.666394 | 2.781623e-22 |
| GDPPER_CO2EMI | 0.654958 | 2.450029e-21 |
| TUBINC_DTHCMD | 0.643615 | 1.936081e-20 |
| URBPOP_LIFEXP | 0.623997 | 5.669778e-19 |
| LIFEXP_EPISCO | 0.620271 | 1.048393e-18 |
| URBPOP_GDPCAP | 0.559181 | 8.624533e-15 |
| CO2EMI_GDPCAP | 0.550221 | 2.782997e-14 |
| URBPOP_CO2EMI | 0.550046 | 2.846393e-14 |
| LIFEXP_CO2EMI | 0.531305 | 2.951829e-13 |
| URBPOP_EPISCO | 0.510131 | 3.507463e-12 |
| POPGRO_TUBINC | 0.442339 | 3.384403e-09 |
| DTHCMD_PM2EXP | 0.283199 | 2.491837e-04 |
| CO2EMI_EPISCO | 0.282734 | 2.553620e-04 |
##################################
# Plotting the correlation matrix
# for all pairwise combinations
# of numeric columns
##################################
cancer_rate_imputed_numeric_correlation = cancer_rate_imputed_numeric.corr()
mask = np.triu(cancer_rate_imputed_numeric_correlation)
plot_correlation_matrix(cancer_rate_imputed_numeric_correlation,mask)
plt.show()
##################################
# Formulating a function
# to plot the correlation matrix
# for all pairwise combinations
# of numeric columns
# with significant p-values only
##################################
def correlation_significance(df=None):
p_matrix = np.zeros(shape=(df.shape[1],df.shape[1]))
for col in df.columns:
for col2 in df.drop(col,axis=1).columns:
_ , p = stats.pearsonr(df[col],df[col2])
p_matrix[df.columns.to_list().index(col),df.columns.to_list().index(col2)] = p
return p_matrix
##################################
# Plotting the correlation matrix
# for all pairwise combinations
# of numeric columns
# with significant p-values only
##################################
cancer_rate_imputed_numeric_correlation_p_values = correlation_significance(cancer_rate_imputed_numeric)
mask = np.invert(np.tril(cancer_rate_imputed_numeric_correlation_p_values<0.05))
plot_correlation_matrix(cancer_rate_imputed_numeric_correlation,mask)
##################################
# Filtering out one among the
# highly correlated variable pairs with
# lesser Pearson.Correlation.Coefficient
# when compared to the target variable
##################################
cancer_rate_imputed_numeric.drop(['GDPPER','METEMI'], inplace=True, axis=1)
##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_imputed_numeric.shape)
Dataset Dimensions:
(163, 13)
Yeo-Johnson Transformation applies a new family of distributions that can be used without restrictions, extending many of the good properties of the Box-Cox power family. Similar to the Box-Cox transformation, the method also estimates the optimal value of lambda but has the ability to transform both positive and negative values by inflating low variance data and deflating high variance data to create a more uniform data set. While there are no restrictions in terms of the applicable values, the interpretability of the transformed values is more diminished as compared to the other methods.
##################################
# Conducting a Yeo-Johnson Transformation
# to address the distributional
# shape of the variables
##################################
yeo_johnson_transformer = PowerTransformer(method='yeo-johnson',
standardize=False)
cancer_rate_imputed_numeric_array = yeo_johnson_transformer.fit_transform(cancer_rate_imputed_numeric)
##################################
# Formulating a new dataset object
# for the transformed data
##################################
cancer_rate_transformed_numeric = pd.DataFrame(cancer_rate_imputed_numeric_array,
columns=cancer_rate_imputed_numeric.columns)
##################################
# Formulating the individual boxplots
# for all transformed numeric columns
##################################
for column in cancer_rate_transformed_numeric:
plt.figure(figsize=(17,1))
sns.boxplot(data=cancer_rate_transformed_numeric, x=column)
##################################
# Filtering out the column
# which remained skewed even
# after applying shape transformation
##################################
cancer_rate_transformed_numeric.drop(['PM2EXP'], inplace=True, axis=1)
##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_transformed_numeric.shape)
Dataset Dimensions:
(163, 12)
##################################
# Conducting standardization
# to transform the values of the
# variables into comparable scale
##################################
standardization_scaler = StandardScaler()
cancer_rate_transformed_numeric_array = standardization_scaler.fit_transform(cancer_rate_transformed_numeric)
##################################
# Formulating a new dataset object
# for the scaled data
##################################
cancer_rate_scaled_numeric = pd.DataFrame(cancer_rate_transformed_numeric_array,
columns=cancer_rate_transformed_numeric.columns)
##################################
# Formulating the individual boxplots
# for all transformed numeric columns
##################################
for column in cancer_rate_scaled_numeric:
plt.figure(figsize=(17,1))
sns.boxplot(data=cancer_rate_scaled_numeric, x=column)
##################################
# Formulating the categorical column
# for encoding transformation
##################################
cancer_rate_categorical_encoded = pd.DataFrame(cancer_rate_cleaned_categorical.loc[:, 'HDICAT'].to_list(),
columns=['HDICAT'])
##################################
# Applying a one-hot encoding transformation
# for the categorical column
##################################
cancer_rate_categorical_encoded = pd.get_dummies(cancer_rate_categorical_encoded, columns=['HDICAT'])
##################################
# Consolidating both numeric columns
# and encoded categorical columns
##################################
cancer_rate_preprocessed = pd.concat([cancer_rate_scaled_numeric,cancer_rate_categorical_encoded], axis=1, join='inner')
##################################
# Performing a general exploration of the consolidated dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_preprocessed.shape)
Dataset Dimensions:
(163, 16)
##################################
# Segregating the target
# and predictor variable lists
##################################
cancer_rate_preprocessed_target = cancer_rate_filtered_row['CANRAT'].to_frame()
cancer_rate_preprocessed_target.reset_index(inplace=True, drop=True)
cancer_rate_preprocessed_categorical = cancer_rate_preprocessed[cancer_rate_categorical_encoded.columns]
cancer_rate_preprocessed_categorical_combined = cancer_rate_preprocessed_categorical.join(cancer_rate_preprocessed_target)
cancer_rate_preprocessed = cancer_rate_preprocessed.drop(cancer_rate_categorical_encoded.columns, axis=1)
cancer_rate_preprocessed_predictors = cancer_rate_preprocessed.columns
cancer_rate_preprocessed_combined = cancer_rate_preprocessed.join(cancer_rate_preprocessed_target)
cancer_rate_preprocessed_all = cancer_rate_preprocessed_combined.join(cancer_rate_categorical_encoded)
##################################
# Segregating the target
# and predictor variable names
##################################
y_variable = 'CANRAT'
x_variables = cancer_rate_preprocessed_predictors
##################################
# Defining the number of
# rows and columns for the subplots
##################################
num_rows = 6
num_cols = 2
##################################
# Formulating the subplot structure
##################################
fig, axes = plt.subplots(num_rows, num_cols, figsize=(15, 30))
##################################
# Flattening the multi-row and
# multi-column axes
##################################
axes = axes.ravel()
##################################
# Formulating the individual boxplots
# for all scaled numeric columns
##################################
for i, x_variable in enumerate(x_variables):
ax = axes[i]
ax.boxplot([group[x_variable] for name, group in cancer_rate_preprocessed_combined.groupby(y_variable)])
ax.set_title(f'{y_variable} Versus {x_variable}')
ax.set_xlabel(y_variable)
ax.set_ylabel(x_variable)
ax.set_xticks(range(1, len(cancer_rate_preprocessed_combined[y_variable].unique()) + 1), ['Low', 'High'])
##################################
# Adjusting the subplot layout
##################################
plt.tight_layout()
##################################
# Presenting the subplots
##################################
plt.show()
##################################
# Segregating the target
# and predictor variable names
##################################
y_variables = cancer_rate_preprocessed_categorical.columns
x_variable = 'CANRAT'
##################################
# Defining the number of
# rows and columns for the subplots
##################################
num_rows = 2
num_cols = 2
##################################
# Formulating the subplot structure
##################################
fig, axes = plt.subplots(num_rows, num_cols, figsize=(15, 10))
##################################
# Flattening the multi-row and
# multi-column axes
##################################
axes = axes.ravel()
##################################
# Formulating the individual stacked column plots
# for all categorical columns
##################################
for i, y_variable in enumerate(y_variables):
ax = axes[i]
category_counts = cancer_rate_preprocessed_categorical_combined.groupby([x_variable, y_variable]).size().unstack(fill_value=0)
category_proportions = category_counts.div(category_counts.sum(axis=1), axis=0)
category_proportions.plot(kind='bar', stacked=True, ax=ax)
ax.set_title(f'{x_variable} Versus {y_variable}')
ax.set_xlabel(x_variable)
ax.set_ylabel('Proportions')
##################################
# Adjusting the subplot layout
##################################
plt.tight_layout()
##################################
# Presenting the subplots
##################################
plt.show()
##################################
# Computing the t-test
# statistic and p-values
# between the target variable
# and numeric predictor columns
##################################
cancer_rate_preprocessed_numeric_ttest_target = {}
cancer_rate_preprocessed_numeric = cancer_rate_preprocessed_combined
cancer_rate_preprocessed_numeric_columns = cancer_rate_preprocessed_predictors
for numeric_column in cancer_rate_preprocessed_numeric_columns:
group_0 = cancer_rate_preprocessed_numeric[cancer_rate_preprocessed_numeric.loc[:,'CANRAT']=='Low']
group_1 = cancer_rate_preprocessed_numeric[cancer_rate_preprocessed_numeric.loc[:,'CANRAT']=='High']
cancer_rate_preprocessed_numeric_ttest_target['CANRAT_' + numeric_column] = stats.ttest_ind(
group_0[numeric_column],
group_1[numeric_column],
equal_var=True)
##################################
# Formulating the pairwise ttest summary
# between the target variable
# and numeric predictor columns
##################################
cancer_rate_preprocessed_numeric_summary = cancer_rate_preprocessed_numeric.from_dict(cancer_rate_preprocessed_numeric_ttest_target, orient='index')
cancer_rate_preprocessed_numeric_summary.columns = ['T.Test.Statistic', 'T.Test.PValue']
display(cancer_rate_preprocessed_numeric_summary.sort_values(by=['T.Test.PValue'], ascending=True).head(12))
| T.Test.Statistic | T.Test.PValue | |
|---|---|---|
| CANRAT_GDPCAP | -11.936988 | 6.247937e-24 |
| CANRAT_EPISCO | -11.788870 | 1.605980e-23 |
| CANRAT_LIFEXP | -10.979098 | 2.754214e-21 |
| CANRAT_TUBINC | 9.608760 | 1.463678e-17 |
| CANRAT_DTHCMD | 8.375558 | 2.552108e-14 |
| CANRAT_CO2EMI | -7.030702 | 5.537463e-11 |
| CANRAT_URBPOP | -6.541001 | 7.734940e-10 |
| CANRAT_POPGRO | 4.904817 | 2.269446e-06 |
| CANRAT_GHGEMI | -2.243089 | 2.625563e-02 |
| CANRAT_FORARE | -1.174143 | 2.420717e-01 |
| CANRAT_POPDEN | -0.495221 | 6.211191e-01 |
| CANRAT_AGRLND | -0.047628 | 9.620720e-01 |
##################################
# Computing the chisquare
# statistic and p-values
# between the target variable
# and categorical predictor columns
##################################
cancer_rate_preprocessed_categorical_chisquare_target = {}
cancer_rate_preprocessed_categorical = cancer_rate_preprocessed_categorical_combined
cancer_rate_preprocessed_categorical_columns = ['HDICAT_L','HDICAT_M','HDICAT_H','HDICAT_VH']
for categorical_column in cancer_rate_preprocessed_categorical_columns:
contingency_table = pd.crosstab(cancer_rate_preprocessed_categorical[categorical_column],
cancer_rate_preprocessed_categorical['CANRAT'])
cancer_rate_preprocessed_categorical_chisquare_target['CANRAT_' + categorical_column] = stats.chi2_contingency(
contingency_table)[0:2]
##################################
# Formulating the pairwise chisquare summary
# between the target variable
# and categorical predictor columns
##################################
cancer_rate_preprocessed_categorical_summary = cancer_rate_preprocessed_categorical.from_dict(cancer_rate_preprocessed_categorical_chisquare_target, orient='index')
cancer_rate_preprocessed_categorical_summary.columns = ['ChiSquare.Test.Statistic', 'ChiSquare.Test.PValue']
display(cancer_rate_preprocessed_categorical_summary.sort_values(by=['ChiSquare.Test.PValue'], ascending=True).head(4))
| ChiSquare.Test.Statistic | ChiSquare.Test.PValue | |
|---|---|---|
| CANRAT_HDICAT_VH | 76.764134 | 1.926446e-18 |
| CANRAT_HDICAT_M | 13.860367 | 1.969074e-04 |
| CANRAT_HDICAT_L | 10.285575 | 1.340742e-03 |
| CANRAT_HDICAT_H | 9.080788 | 2.583087e-03 |
Hyperparameter tuning is an iterative process that involves experimenting with different hyperparameter combinations, evaluating the model's performance, and refining the hyperparameter values to achieve the best possible performance on new, unseen data - aimed at building effective and well-generalizing machine learning models. A model's performance depends not only on the learned parameters (weights) during training but also on hyperparameters, which are external configuration settings that cannot be learned from the data.
##################################
# Consolidating relevant numeric columns
# and encoded categorical columns
# after hypothesis testing
##################################
cancer_rate_premodelling = cancer_rate_preprocessed_all.drop(['AGRLND','POPDEN','GHGEMI','POPGRO','FORARE','HDICAT_H','HDICAT_M','HDICAT_L'], axis=1)
##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_premodelling.shape)
Dataset Dimensions:
(163, 9)
##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate_premodelling.dtypes)
Column Names and Data Types:
URBPOP float64 LIFEXP float64 TUBINC float64 DTHCMD float64 CO2EMI float64 GDPCAP float64 EPISCO float64 CANRAT category HDICAT_VH uint8 dtype: object
##################################
# Gathering the pairplot for all variables
##################################
sns.pairplot(cancer_rate_premodelling, kind='reg')
plt.show()
##################################
# Separating the target
# and predictor columns
##################################
X = cancer_rate_premodelling.drop('CANRAT', axis = 1)
y = cancer_rate_premodelling['CANRAT'].cat.codes
##################################
# Formulating the train and test data
# using a 70-30 ratio
##################################
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.3, random_state= 88888888, stratify=y)
##################################
# Performing a general exploration of the train dataset
##################################
print('Dataset Dimensions: ')
display(X_train.shape)
Dataset Dimensions:
(114, 8)
##################################
# Validating the class distribution of the train dataset
##################################
y_train.value_counts(normalize = True)
0 0.745614 1 0.254386 dtype: float64
##################################
# Performing a general exploration of the test dataset
##################################
print('Dataset Dimensions: ')
display(X_test.shape)
Dataset Dimensions:
(49, 8)
##################################
# Validating the class distribution of the test dataset
##################################
y_test.value_counts(normalize = True)
0 0.755102 1 0.244898 dtype: float64
##################################
# Defining a function to compute
# model performance
##################################
def model_performance_evaluation(y_true, y_pred):
metric_name = ['Accuracy','Precision','Recall','F1','AUROC']
metric_value = [accuracy_score(y_true, y_pred),
precision_score(y_true, y_pred),
recall_score(y_true, y_pred),
f1_score(y_true, y_pred),
roc_auc_score(y_true, y_pred)]
metric_summary = pd.DataFrame(zip(metric_name, metric_value),
columns=['metric_name','metric_value'])
return(metric_summary)
Logistic Regression models the relationship between the probability of an event (among two outcome levels) by having the log-odds of the event be a linear combination of a set of predictors weighted by their respective parameter estimates. The parameters are estimated via maximum likelihood estimation by testing different values through multiple iterations to optimize for the best fit of log odds. All of these iterations produce the log likelihood function, and logistic regression seeks to maximize this function to find the best parameter estimates. Given the optimal parameters, the conditional probabilities for each observation can be calculated, logged, and summed together to yield a predicted probability.
##################################
# Creating an instance of the
# Logistic Regression model
##################################
logistic_regression = LogisticRegression()
##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
hyperparameter_grid = {
'C': [1.0],
'penalty': ['l1', 'l2'],
'solver': ['liblinear','saga'],
'class_weight': [None],
'random_state': [88888888]}
##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
optimal_logistic_regression = GridSearchCV(estimator = logistic_regression,
param_grid = hyperparameter_grid,
n_jobs = -1,
scoring='f1')
##################################
# Fitting the optimal Logistic Regression model
##################################
optimal_logistic_regression.fit(X_train, y_train)
##################################
# Determining the optimal hyperparameter
# for the Logistic Regression model
##################################
optimal_logistic_regression.best_score_
optimal_logistic_regression.best_params_
{'C': 1.0,
'class_weight': None,
'penalty': 'l1',
'random_state': 88888888,
'solver': 'liblinear'}
##################################
# Evaluating the optimal logistic regression model
# on the train set
##################################
optimal_logistic_regression_y_hat_train = optimal_logistic_regression.predict(X_train)
##################################
# Gathering the model evaluation metrics
##################################
optimal_logistic_regression_performance_train = model_performance_evaluation(y_train, optimal_logistic_regression_y_hat_train)
optimal_logistic_regression_performance_train['model'] = ['optimal_logistic_regression'] * 5
optimal_logistic_regression_performance_train['set'] = ['train'] * 5
print('Optimal Logistic Regression Model Performance on Train Data: ')
display(optimal_logistic_regression_performance_train)
Optimal Logistic Regression Model Performance on Train Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.947368 | optimal_logistic_regression | train |
| 1 | Precision | 0.870968 | optimal_logistic_regression | train |
| 2 | Recall | 0.931034 | optimal_logistic_regression | train |
| 3 | F1 | 0.900000 | optimal_logistic_regression | train |
| 4 | AUROC | 0.941988 | optimal_logistic_regression | train |
##################################
# Evaluating the optimal logistic regression model
# on the test set
##################################
optimal_logistic_regression_y_hat_test = optimal_logistic_regression.predict(X_test)
##################################
# Gathering the model evaluation metrics
##################################
optimal_logistic_regression_performance_test = model_performance_evaluation(y_test, optimal_logistic_regression_y_hat_test)
optimal_logistic_regression_performance_test['model'] = ['optimal_logistic_regression'] * 5
optimal_logistic_regression_performance_test['set'] = ['test'] * 5
print('Optimal Logistic Regression Model Performance on Test Data: ')
display(optimal_logistic_regression_performance_test)
Optimal Logistic Regression Model Performance on Test Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.897959 | optimal_logistic_regression | test |
| 1 | Precision | 0.888889 | optimal_logistic_regression | test |
| 2 | Recall | 0.666667 | optimal_logistic_regression | test |
| 3 | F1 | 0.761905 | optimal_logistic_regression | test |
| 4 | AUROC | 0.819820 | optimal_logistic_regression | test |
Decision trees create a model that predicts the class label of a sample based on input features. A decision tree consists of nodes that represent decisions or choices, edges which connect nodes and represent the possible outcomes of a decision and leaf (or terminal) nodes which represent the final decision or the predicted class label. The decision-making process involves feature selection (at each internal node, the algorithm decides which feature to split on based on a certain criterion including gini impurity or entropy), splitting criteria (the splitting criteria aim to find the feature and its corresponding threshold that best separates the data into different classes. The goal is to increase homogeneity within each resulting subset), recursive splitting (the process of feature selection and splitting continues recursively, creating a tree structure. The dataset is partitioned at each internal node based on the chosen feature, and the process repeats for each subset) and stopping criteria (the recursion stops when a certain condition is met, known as a stopping criterion. Common stopping criteria include a maximum depth for the tree, a minimum number of samples required to split a node, or a minimum number of samples in a leaf node.)
##################################
# Creating an instance of the
# Decision Tree model
##################################
decision_tree = DecisionTreeClassifier()
##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
hyperparameter_grid = {
'criterion': ['gini','entropy','log_loss'],
'max_depth': [3,5,7],
'min_samples_leaf': [3,5,10],
'class_weight': [None],
'random_state': [88888888]}
##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
optimal_decision_tree = GridSearchCV(estimator = decision_tree,
param_grid = hyperparameter_grid,
n_jobs = -1,
scoring='f1')
##################################
# Fitting the optimal Decision Tree model
##################################
optimal_decision_tree.fit(X_train, y_train)
##################################
# Determining the optimal hyperparameter
# for the Decision Tree model
##################################
optimal_decision_tree.best_score_
optimal_decision_tree.best_params_
{'class_weight': None,
'criterion': 'entropy',
'max_depth': 5,
'min_samples_leaf': 3,
'random_state': 88888888}
##################################
# Evaluating the optimal decision tree model
# on the train set
##################################
optimal_decision_tree_y_hat_train = optimal_decision_tree.predict(X_train)
##################################
# Gathering the model evaluation metrics
##################################
optimal_decision_tree_performance_train = model_performance_evaluation(y_train, optimal_decision_tree_y_hat_train)
optimal_decision_tree_performance_train['model'] = ['optimal_decision_tree'] * 5
optimal_decision_tree_performance_train['set'] = ['train'] * 5
print('Optimal Decision Tree Model Performance on Train Data: ')
display(optimal_decision_tree_performance_train)
Optimal Decision Tree Model Performance on Train Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.973684 | optimal_decision_tree | train |
| 1 | Precision | 1.000000 | optimal_decision_tree | train |
| 2 | Recall | 0.896552 | optimal_decision_tree | train |
| 3 | F1 | 0.945455 | optimal_decision_tree | train |
| 4 | AUROC | 0.948276 | optimal_decision_tree | train |
##################################
# Evaluating the optimal decision tree model
# on the test set
##################################
optimal_decision_tree_y_hat_test = optimal_decision_tree.predict(X_test)
##################################
# Gathering the model evaluation metrics
##################################
optimal_decision_tree_performance_test = model_performance_evaluation(y_test, optimal_decision_tree_y_hat_test)
optimal_decision_tree_performance_test['model'] = ['optimal_decision_tree'] * 5
optimal_decision_tree_performance_test['set'] = ['test'] * 5
print('Optimal Decision Tree Model Performance on Test Data: ')
display(optimal_decision_tree_performance_test)
Optimal Decision Tree Model Performance on Test Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.857143 | optimal_decision_tree | test |
| 1 | Precision | 0.857143 | optimal_decision_tree | test |
| 2 | Recall | 0.500000 | optimal_decision_tree | test |
| 3 | F1 | 0.631579 | optimal_decision_tree | test |
| 4 | AUROC | 0.736486 | optimal_decision_tree | test |
Random Forest is an ensemble learning method made up of a large set of small decision trees called estimators, with each producing its own prediction. The random forest model aggregates the predictions of the estimators to produce a more accurate prediction. The algorithm involves bootstrap aggregating (where smaller subsets of the training data are repeatedly subsampled with replacement), random subspacing (where a subset of features are sampled and used to train each individual estimator), estimator training (where unpruned decision trees are formulated for each estimator) and inference by aggregating the predictions of all estimators.
##################################
# Creating an instance of the
# Random Forest model
##################################
random_forest = RandomForestClassifier()
##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
hyperparameter_grid = {
'criterion': ['gini','entropy','log_loss'],
'max_depth': [3,5,7],
'min_samples_leaf': [3,5,10],
'n_estimators': [3,5,7],
'max_features':['sqrt', 'log2'],
'class_weight': [None],
'random_state': [88888888]}
##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
optimal_random_forest = GridSearchCV(estimator = random_forest,
param_grid = hyperparameter_grid,
n_jobs = -1,
scoring='f1')
##################################
# Fitting the optimal Random Forest model
##################################
optimal_random_forest.fit(X_train, y_train)
##################################
# Determining the optimal hyperparameter
# for the Random Forest model
##################################
optimal_random_forest.best_score_
optimal_random_forest.best_params_
{'class_weight': None,
'criterion': 'gini',
'max_depth': 5,
'max_features': 'log2',
'min_samples_leaf': 3,
'n_estimators': 7,
'random_state': 88888888}
##################################
# Evaluating the optimal Random Forest model
# on the train set
##################################
optimal_random_forest_y_hat_train = optimal_random_forest.predict(X_train)
##################################
# Gathering the model evaluation metrics
##################################
optimal_random_forest_performance_train = model_performance_evaluation(y_train, optimal_random_forest_y_hat_train)
optimal_random_forest_performance_train['model'] = ['optimal_random_forest'] * 5
optimal_random_forest_performance_train['set'] = ['train'] * 5
print('Optimal Random Forest Model Performance on Train Data: ')
display(optimal_random_forest_performance_train)
Optimal Random Forest Model Performance on Train Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.964912 | optimal_random_forest | train |
| 1 | Precision | 0.931034 | optimal_random_forest | train |
| 2 | Recall | 0.931034 | optimal_random_forest | train |
| 3 | F1 | 0.931034 | optimal_random_forest | train |
| 4 | AUROC | 0.953753 | optimal_random_forest | train |
##################################
# Evaluating the optimal Random Forest model
# on the test set
##################################
optimal_random_forest_y_hat_test = optimal_random_forest.predict(X_test)
##################################
# Gathering the model evaluation metrics
##################################
optimal_random_forest_performance_test = model_performance_evaluation(y_test, optimal_random_forest_y_hat_test)
optimal_random_forest_performance_test['model'] = ['optimal_random_forest'] * 5
optimal_random_forest_performance_test['set'] = ['test'] * 5
print('Optimal Random Forest Model Performance on Test Data: ')
display(optimal_random_forest_performance_test)
Optimal Random Forest Model Performance on Test Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.897959 | optimal_random_forest | test |
| 1 | Precision | 0.888889 | optimal_random_forest | test |
| 2 | Recall | 0.666667 | optimal_random_forest | test |
| 3 | F1 | 0.761905 | optimal_random_forest | test |
| 4 | AUROC | 0.819820 | optimal_random_forest | test |
Support Vector Machine plots each observation in an N-dimensional space corresponding to the number of features in the data set and finds a hyperplane that maximally separates the different classes by a maximally large margin (which is defined as the distance between the hyperplane and the closest data points from each class). The algorithm applies kernel transformation by mapping non-linearly separable data using the similarities between the points in a high-dimensional feature space for improved discrimination.
##################################
# Creating an instance of the
# Support Vector Machine model
##################################
support_vector_machine = SVC()
##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
hyperparameter_grid = {
'C': [1.0],
'kernel': ['linear', 'poly', 'rbf', 'sigmoid'],
'class_weight': [None],
'random_state': [88888888]}
##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
optimal_support_vector_machine = GridSearchCV(estimator = support_vector_machine,
param_grid = hyperparameter_grid,
n_jobs = -1,
scoring='f1')
##################################
# Fitting the optimal Support Vector Machine model
##################################
optimal_support_vector_machine.fit(X_train, y_train)
##################################
# Determining the optimal hyperparameter
# for the Support Vector Machine model
##################################
optimal_support_vector_machine.best_score_
optimal_support_vector_machine.best_params_
{'C': 1.0, 'class_weight': None, 'kernel': 'poly', 'random_state': 88888888}
##################################
# Evaluating the optimal Support Vector Machine model
# on the train set
##################################
optimal_support_vector_machine_y_hat_train = optimal_support_vector_machine.predict(X_train)
##################################
# Gathering the model evaluation metrics
##################################
optimal_support_vector_machine_performance_train = model_performance_evaluation(y_train, optimal_support_vector_machine_y_hat_train)
optimal_support_vector_machine_performance_train['model'] = ['optimal_support_vector_machine'] * 5
optimal_support_vector_machine_performance_train['set'] = ['train'] * 5
print('Optimal Support Vector Machine Model Performance on Train Data: ')
display(optimal_support_vector_machine_performance_train)
Optimal Support Vector Machine Model Performance on Train Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.947368 | optimal_support_vector_machine | train |
| 1 | Precision | 0.960000 | optimal_support_vector_machine | train |
| 2 | Recall | 0.827586 | optimal_support_vector_machine | train |
| 3 | F1 | 0.888889 | optimal_support_vector_machine | train |
| 4 | AUROC | 0.907911 | optimal_support_vector_machine | train |
##################################
# Evaluating the optimal Support Vector Machine model
# on the test set
##################################
optimal_support_vector_machine_y_hat_test = optimal_support_vector_machine.predict(X_test)
##################################
# Gathering the model evaluation metrics
##################################
optimal_support_vector_machine_performance_test = model_performance_evaluation(y_test, optimal_support_vector_machine_y_hat_test)
optimal_support_vector_machine_performance_test['model'] = ['optimal_support_vector_machine'] * 5
optimal_support_vector_machine_performance_test['set'] = ['test'] * 5
print('Optimal Support Vector Machine Model Performance on Test Data: ')
display(optimal_support_vector_machine_performance_test)
Optimal Support Vector Machine Model Performance on Test Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.857143 | optimal_support_vector_machine | test |
| 1 | Precision | 0.857143 | optimal_support_vector_machine | test |
| 2 | Recall | 0.500000 | optimal_support_vector_machine | test |
| 3 | F1 | 0.631579 | optimal_support_vector_machine | test |
| 4 | AUROC | 0.736486 | optimal_support_vector_machine | test |
Class weights are used to assign different levels of importance to different classes when the distribution of instances across different classes in a classification problem is not equal. By assigning higher weights to the minority class, the model is encouraged to give more attention to correctly predicting instances from the minority class. Class weights are incorporated into the loss function during training. The loss for each instance is multiplied by its corresponding class weight. This means that misclassifying an instance from the minority class will have a greater impact on the overall loss than misclassifying an instance from the majority class. The use of class weights helps balance the influence of each class during training, mitigating the impact of class imbalance. It provides a way to focus the learning process on the classes that are underrepresented in the training data.
##################################
# Consolidating relevant numeric columns
# and encoded categorical columns
# after hypothesis testing
##################################
cancer_rate_premodelling = cancer_rate_preprocessed_all.drop(['AGRLND','POPDEN','GHGEMI','POPGRO','FORARE','HDICAT_H','HDICAT_M','HDICAT_L'], axis=1)
##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_premodelling.shape)
Dataset Dimensions:
(163, 9)
##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate_premodelling.dtypes)
Column Names and Data Types:
URBPOP float64 LIFEXP float64 TUBINC float64 DTHCMD float64 CO2EMI float64 GDPCAP float64 EPISCO float64 CANRAT category HDICAT_VH uint8 dtype: object
##################################
# Gathering the pairplot for all variables
##################################
sns.pairplot(cancer_rate_premodelling, kind='reg')
plt.show()
##################################
# Separating the target
# and predictor columns
##################################
X = cancer_rate_premodelling.drop('CANRAT', axis = 1)
y = cancer_rate_premodelling['CANRAT'].cat.codes
##################################
# Formulating the train and test data
# using a 70-30 ratio
##################################
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.3, random_state= 88888888, stratify=y)
##################################
# Performing a general exploration of the train dataset
##################################
print('Dataset Dimensions: ')
display(X_train.shape)
Dataset Dimensions:
(114, 8)
##################################
# Validating the class distribution of the train dataset
##################################
y_train.value_counts(normalize = True)
0 0.745614 1 0.254386 dtype: float64
##################################
# Performing a general exploration of the test dataset
##################################
print('Dataset Dimensions: ')
display(X_test.shape)
Dataset Dimensions:
(49, 8)
##################################
# Validating the class distribution of the test dataset
##################################
y_test.value_counts(normalize = True)
0 0.755102 1 0.244898 dtype: float64
##################################
# Defining a function to compute
# model performance
##################################
def model_performance_evaluation(y_true, y_pred):
metric_name = ['Accuracy','Precision','Recall','F1','AUROC']
metric_value = [accuracy_score(y_true, y_pred),
precision_score(y_true, y_pred),
recall_score(y_true, y_pred),
f1_score(y_true, y_pred),
roc_auc_score(y_true, y_pred)]
metric_summary = pd.DataFrame(zip(metric_name, metric_value),
columns=['metric_name','metric_value'])
return(metric_summary)
Logistic Regression models the relationship between the probability of an event (among two outcome levels) by having the log-odds of the event be a linear combination of a set of predictors weighted by their respective parameter estimates. The parameters are estimated via maximum likelihood estimation by testing different values through multiple iterations to optimize for the best fit of log odds. All of these iterations produce the log likelihood function, and logistic regression seeks to maximize this function to find the best parameter estimates. Given the optimal parameters, the conditional probabilities for each observation can be calculated, logged, and summed together to yield a predicted probability.
##################################
# Creating an instance of the
# Logistic Regression model
##################################
logistic_regression = LogisticRegression()
##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
hyperparameter_grid = {
'C': [1.0],
'penalty': ['l1', 'l2'],
'solver': ['liblinear','saga'],
'class_weight': [{0:0.25, 1:0.75}],
'random_state': [88888888]}
##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
weighted_logistic_regression = GridSearchCV(estimator = logistic_regression,
param_grid = hyperparameter_grid,
n_jobs = -1,
scoring='f1')
##################################
# Fitting the weighted Logistic Regression model
##################################
weighted_logistic_regression.fit(X_train, y_train)
##################################
# Determining the weighted hyperparameter
# for the Logistic Regression model
##################################
weighted_logistic_regression.best_score_
weighted_logistic_regression.best_params_
{'C': 1.0,
'class_weight': {0: 0.25, 1: 0.75},
'penalty': 'l2',
'random_state': 88888888,
'solver': 'liblinear'}
##################################
# Evaluating the weighted logistic regression model
# on the train set
##################################
weighted_logistic_regression_y_hat_train = weighted_logistic_regression.predict(X_train)
##################################
# Gathering the model evaluation metrics
##################################
weighted_logistic_regression_performance_train = model_performance_evaluation(y_train, weighted_logistic_regression_y_hat_train)
weighted_logistic_regression_performance_train['model'] = ['weighted_logistic_regression'] * 5
weighted_logistic_regression_performance_train['set'] = ['train'] * 5
print('Weighted Logistic Regression Model Performance on Train Data: ')
display(weighted_logistic_regression_performance_train)
Weighted Logistic Regression Model Performance on Train Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.894737 | weighted_logistic_regression | train |
| 1 | Precision | 0.707317 | weighted_logistic_regression | train |
| 2 | Recall | 1.000000 | weighted_logistic_regression | train |
| 3 | F1 | 0.828571 | weighted_logistic_regression | train |
| 4 | AUROC | 0.929412 | weighted_logistic_regression | train |
##################################
# Evaluating the weighted logistic regression model
# on the test set
##################################
weighted_logistic_regression_y_hat_test = weighted_logistic_regression.predict(X_test)
##################################
# Gathering the model evaluation metrics
##################################
weighted_logistic_regression_performance_test = model_performance_evaluation(y_test, weighted_logistic_regression_y_hat_test)
weighted_logistic_regression_performance_test['model'] = ['weighted_logistic_regression'] * 5
weighted_logistic_regression_performance_test['set'] = ['test'] * 5
print('Weighted Logistic Regression Model Performance on Test Data: ')
display(weighted_logistic_regression_performance_test)
Weighted Logistic Regression Model Performance on Test Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.938776 | weighted_logistic_regression | test |
| 1 | Precision | 0.846154 | weighted_logistic_regression | test |
| 2 | Recall | 0.916667 | weighted_logistic_regression | test |
| 3 | F1 | 0.880000 | weighted_logistic_regression | test |
| 4 | AUROC | 0.931306 | weighted_logistic_regression | test |
Decision trees create a model that predicts the class label of a sample based on input features. A decision tree consists of nodes that represent decisions or choices, edges which connect nodes and represent the possible outcomes of a decision and leaf (or terminal) nodes which represent the final decision or the predicted class label. The decision-making process involves feature selection (at each internal node, the algorithm decides which feature to split on based on a certain criterion including gini impurity or entropy), splitting criteria (the splitting criteria aim to find the feature and its corresponding threshold that best separates the data into different classes. The goal is to increase homogeneity within each resulting subset), recursive splitting (the process of feature selection and splitting continues recursively, creating a tree structure. The dataset is partitioned at each internal node based on the chosen feature, and the process repeats for each subset) and stopping criteria (the recursion stops when a certain condition is met, known as a stopping criterion. Common stopping criteria include a maximum depth for the tree, a minimum number of samples required to split a node, or a minimum number of samples in a leaf node.)
##################################
# Creating an instance of the
# Decision Tree model
##################################
decision_tree = DecisionTreeClassifier()
##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
hyperparameter_grid = {
'criterion': ['gini','entropy','log_loss'],
'max_depth': [3,5,7],
'min_samples_leaf': [3,5,10],
'class_weight': [{0:0.25, 1:0.75}],
'random_state': [88888888]}
##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
weighted_decision_tree = GridSearchCV(estimator = decision_tree,
param_grid = hyperparameter_grid,
n_jobs = -1,
scoring='f1')
##################################
# Fitting the weighted Decision Tree model
##################################
weighted_decision_tree.fit(X_train, y_train)
##################################
# Determining the weighted hyperparameter
# for the Decision Tree model
##################################
weighted_decision_tree.best_score_
weighted_decision_tree.best_params_
{'class_weight': {0: 0.25, 1: 0.75},
'criterion': 'gini',
'max_depth': 3,
'min_samples_leaf': 3,
'random_state': 88888888}
##################################
# Evaluating the weighted decision tree model
# on the train set
##################################
weighted_decision_tree_y_hat_train = weighted_decision_tree.predict(X_train)
##################################
# Gathering the model evaluation metrics
##################################
weighted_decision_tree_performance_train = model_performance_evaluation(y_train, weighted_decision_tree_y_hat_train)
weighted_decision_tree_performance_train['model'] = ['weighted_decision_tree'] * 5
weighted_decision_tree_performance_train['set'] = ['train'] * 5
print('Weighted Decision Tree Model Performance on Train Data: ')
display(weighted_decision_tree_performance_train)
Weighted Decision Tree Model Performance on Train Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.956140 | weighted_decision_tree | train |
| 1 | Precision | 0.852941 | weighted_decision_tree | train |
| 2 | Recall | 1.000000 | weighted_decision_tree | train |
| 3 | F1 | 0.920635 | weighted_decision_tree | train |
| 4 | AUROC | 0.970588 | weighted_decision_tree | train |
##################################
# Evaluating the weighted decision tree model
# on the test set
##################################
weighted_decision_tree_y_hat_test = weighted_decision_tree.predict(X_test)
##################################
# Gathering the model evaluation metrics
##################################
weighted_decision_tree_performance_test = model_performance_evaluation(y_test, weighted_decision_tree_y_hat_test)
weighted_decision_tree_performance_test['model'] = ['weighted_decision_tree'] * 5
weighted_decision_tree_performance_test['set'] = ['test'] * 5
print('Weighted Decision Tree Model Performance on Test Data: ')
display(weighted_decision_tree_performance_test)
Weighted Decision Tree Model Performance on Test Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.897959 | weighted_decision_tree | test |
| 1 | Precision | 0.769231 | weighted_decision_tree | test |
| 2 | Recall | 0.833333 | weighted_decision_tree | test |
| 3 | F1 | 0.800000 | weighted_decision_tree | test |
| 4 | AUROC | 0.876126 | weighted_decision_tree | test |
Random Forest is an ensemble learning method made up of a large set of small decision trees called estimators, with each producing its own prediction. The random forest model aggregates the predictions of the estimators to produce a more accurate prediction. The algorithm involves bootstrap aggregating (where smaller subsets of the training data are repeatedly subsampled with replacement), random subspacing (where a subset of features are sampled and used to train each individual estimator), estimator training (where unpruned decision trees are formulated for each estimator) and inference by aggregating the predictions of all estimators.
##################################
# Creating an instance of the
# Random Forest model
##################################
random_forest = RandomForestClassifier()
##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
hyperparameter_grid = {
'criterion': ['gini','entropy','log_loss'],
'max_depth': [3,5,7],
'min_samples_leaf': [3,5,10],
'n_estimators': [3,5,7],
'max_features':['sqrt', 'log2'],
'class_weight': [{0:0.25, 1:0.75}],
'random_state': [88888888]}
##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
weighted_random_forest = GridSearchCV(estimator = random_forest,
param_grid = hyperparameter_grid,
n_jobs = -1,
scoring='f1')
##################################
# Fitting the weighted Random Forest model
##################################
weighted_random_forest.fit(X_train, y_train)
##################################
# Determining the weighted hyperparameter
# for the Random Forest model
##################################
weighted_random_forest.best_score_
weighted_random_forest.best_params_
{'class_weight': {0: 0.25, 1: 0.75},
'criterion': 'entropy',
'max_depth': 3,
'max_features': 'sqrt',
'min_samples_leaf': 5,
'n_estimators': 3,
'random_state': 88888888}
##################################
# Evaluating the weighted Random Forest model
# on the train set
##################################
weighted_random_forest_y_hat_train = weighted_random_forest.predict(X_train)
##################################
# Gathering the model evaluation metrics
##################################
weighted_random_forest_performance_train = model_performance_evaluation(y_train, weighted_random_forest_y_hat_train)
weighted_random_forest_performance_train['model'] = ['weighted_random_forest'] * 5
weighted_random_forest_performance_train['set'] = ['train'] * 5
print('Weighted Random Forest Model Performance on Train Data: ')
display(weighted_random_forest_performance_train)
Weighted Random Forest Model Performance on Train Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.929825 | weighted_random_forest | train |
| 1 | Precision | 0.800000 | weighted_random_forest | train |
| 2 | Recall | 0.965517 | weighted_random_forest | train |
| 3 | F1 | 0.875000 | weighted_random_forest | train |
| 4 | AUROC | 0.941582 | weighted_random_forest | train |
##################################
# Evaluating the weighted Random Forest model
# on the test set
##################################
weighted_random_forest_y_hat_test = weighted_random_forest.predict(X_test)
##################################
# Gathering the model evaluation metrics
##################################
weighted_random_forest_performance_test = model_performance_evaluation(y_test, weighted_random_forest_y_hat_test)
weighted_random_forest_performance_test['model'] = ['weighted_random_forest'] * 5
weighted_random_forest_performance_test['set'] = ['test'] * 5
print('Weighted Random Forest Model Performance on Test Data: ')
display(weighted_random_forest_performance_test)
Weighted Random Forest Model Performance on Test Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.938776 | weighted_random_forest | test |
| 1 | Precision | 0.909091 | weighted_random_forest | test |
| 2 | Recall | 0.833333 | weighted_random_forest | test |
| 3 | F1 | 0.869565 | weighted_random_forest | test |
| 4 | AUROC | 0.903153 | weighted_random_forest | test |
Support Vector Machine plots each observation in an N-dimensional space corresponding to the number of features in the data set and finds a hyperplane that maximally separates the different classes by a maximally large margin (which is defined as the distance between the hyperplane and the closest data points from each class). The algorithm applies kernel transformation by mapping non-linearly separable data using the similarities between the points in a high-dimensional feature space for improved discrimination.
##################################
# Creating an instance of the
# Support Vector Machine model
##################################
support_vector_machine = SVC()
##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
hyperparameter_grid = {
'C': [1.0],
'kernel': ['linear', 'poly', 'rbf', 'sigmoid'],
'class_weight': [{0:0.25, 1:0.75}],
'random_state': [88888888]}
##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
weighted_support_vector_machine = GridSearchCV(estimator = support_vector_machine,
param_grid = hyperparameter_grid,
n_jobs = -1,
scoring='f1')
##################################
# Fitting the weighted Support Vector Machine model
##################################
weighted_support_vector_machine.fit(X_train, y_train)
##################################
# Determining the weighted hyperparameter
# for the Support Vector Machine model
##################################
weighted_support_vector_machine.best_score_
weighted_support_vector_machine.best_params_
{'C': 1.0,
'class_weight': {0: 0.25, 1: 0.75},
'kernel': 'poly',
'random_state': 88888888}
##################################
# Evaluating the weighted Support Vector Machine model
# on the train set
##################################
weighted_support_vector_machine_y_hat_train = weighted_support_vector_machine.predict(X_train)
##################################
# Gathering the model evaluation metrics
##################################
weighted_support_vector_machine_performance_train = model_performance_evaluation(y_train, weighted_support_vector_machine_y_hat_train)
weighted_support_vector_machine_performance_train['model'] = ['weighted_support_vector_machine'] * 5
weighted_support_vector_machine_performance_train['set'] = ['train'] * 5
print('Weighted Support Vector Machine Model Performance on Train Data: ')
display(weighted_support_vector_machine_performance_train)
Weighted Support Vector Machine Model Performance on Train Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.964912 | weighted_support_vector_machine | train |
| 1 | Precision | 0.962963 | weighted_support_vector_machine | train |
| 2 | Recall | 0.896552 | weighted_support_vector_machine | train |
| 3 | F1 | 0.928571 | weighted_support_vector_machine | train |
| 4 | AUROC | 0.942394 | weighted_support_vector_machine | train |
##################################
# Evaluating the weighted Support Vector Machine model
# on the test set
##################################
weighted_support_vector_machine_y_hat_test = weighted_support_vector_machine.predict(X_test)
##################################
# Gathering the model evaluation metrics
##################################
weighted_support_vector_machine_performance_test = model_performance_evaluation(y_test, weighted_support_vector_machine_y_hat_test)
weighted_support_vector_machine_performance_test['model'] = ['weighted_support_vector_machine'] * 5
weighted_support_vector_machine_performance_test['set'] = ['test'] * 5
print('Weighted Support Vector Machine Model Performance on Test Data: ')
display(weighted_support_vector_machine_performance_test)
Weighted Support Vector Machine Model Performance on Test Data:
| metric_name | metric_value | model | set | |
|---|---|---|---|---|
| 0 | Accuracy | 0.877551 | weighted_support_vector_machine | test |
| 1 | Precision | 0.875000 | weighted_support_vector_machine | test |
| 2 | Recall | 0.583333 | weighted_support_vector_machine | test |
| 3 | F1 | 0.700000 | weighted_support_vector_machine | test |
| 4 | AUROC | 0.778153 | weighted_support_vector_machine | test |
Logistic Regression models the relationship between the probability of an event (among two outcome levels) by having the log-odds of the event be a linear combination of a set of predictors weighted by their respective parameter estimates. The parameters are estimated via maximum likelihood estimation by testing different values through multiple iterations to optimize for the best fit of log odds. All of these iterations produce the log likelihood function, and logistic regression seeks to maximize this function to find the best parameter estimates. Given the optimal parameters, the conditional probabilities for each observation can be calculated, logged, and summed together to yield a predicted probability.
Decision trees create a model that predicts the class label of a sample based on input features. A decision tree consists of nodes that represent decisions or choices, edges which connect nodes and represent the possible outcomes of a decision and leaf (or terminal) nodes which represent the final decision or the predicted class label. The decision-making process involves feature selection (at each internal node, the algorithm decides which feature to split on based on a certain criterion including gini impurity or entropy), splitting criteria (the splitting criteria aim to find the feature and its corresponding threshold that best separates the data into different classes. The goal is to increase homogeneity within each resulting subset), recursive splitting (the process of feature selection and splitting continues recursively, creating a tree structure. The dataset is partitioned at each internal node based on the chosen feature, and the process repeats for each subset) and stopping criteria (the recursion stops when a certain condition is met, known as a stopping criterion. Common stopping criteria include a maximum depth for the tree, a minimum number of samples required to split a node, or a minimum number of samples in a leaf node.)
Random Forest is an ensemble learning method made up of a large set of small decision trees called estimators, with each producing its own prediction. The random forest model aggregates the predictions of the estimators to produce a more accurate prediction. The algorithm involves bootstrap aggregating (where smaller subsets of the training data are repeatedly subsampled with replacement), random subspacing (where a subset of features are sampled and used to train each individual estimator), estimator training (where unpruned decision trees are formulated for each estimator) and inference by aggregating the predictions of all estimators.
Support Vector Machine plots each observation in an N-dimensional space corresponding to the number of features in the data set and finds a hyperplane that maximally separates the different classes by a maximally large margin (which is defined as the distance between the hyperplane and the closest data points from each class). The algorithm applies kernel transformation by mapping non-linearly separable data using the similarities between the points in a high-dimensional feature space for improved discrimination.
Logistic Regression models the relationship between the probability of an event (among two outcome levels) by having the log-odds of the event be a linear combination of a set of predictors weighted by their respective parameter estimates. The parameters are estimated via maximum likelihood estimation by testing different values through multiple iterations to optimize for the best fit of log odds. All of these iterations produce the log likelihood function, and logistic regression seeks to maximize this function to find the best parameter estimates. Given the optimal parameters, the conditional probabilities for each observation can be calculated, logged, and summed together to yield a predicted probability.
Decision trees create a model that predicts the class label of a sample based on input features. A decision tree consists of nodes that represent decisions or choices, edges which connect nodes and represent the possible outcomes of a decision and leaf (or terminal) nodes which represent the final decision or the predicted class label. The decision-making process involves feature selection (at each internal node, the algorithm decides which feature to split on based on a certain criterion including gini impurity or entropy), splitting criteria (the splitting criteria aim to find the feature and its corresponding threshold that best separates the data into different classes. The goal is to increase homogeneity within each resulting subset), recursive splitting (the process of feature selection and splitting continues recursively, creating a tree structure. The dataset is partitioned at each internal node based on the chosen feature, and the process repeats for each subset) and stopping criteria (the recursion stops when a certain condition is met, known as a stopping criterion. Common stopping criteria include a maximum depth for the tree, a minimum number of samples required to split a node, or a minimum number of samples in a leaf node.)
Random Forest is an ensemble learning method made up of a large set of small decision trees called estimators, with each producing its own prediction. The random forest model aggregates the predictions of the estimators to produce a more accurate prediction. The algorithm involves bootstrap aggregating (where smaller subsets of the training data are repeatedly subsampled with replacement), random subspacing (where a subset of features are sampled and used to train each individual estimator), estimator training (where unpruned decision trees are formulated for each estimator) and inference by aggregating the predictions of all estimators.
Support Vector Machine plots each observation in an N-dimensional space corresponding to the number of features in the data set and finds a hyperplane that maximally separates the different classes by a maximally large margin (which is defined as the distance between the hyperplane and the closest data points from each class). The algorithm applies kernel transformation by mapping non-linearly separable data using the similarities between the points in a high-dimensional feature space for improved discrimination.
Logistic Regression models the relationship between the probability of an event (among two outcome levels) by having the log-odds of the event be a linear combination of a set of predictors weighted by their respective parameter estimates. The parameters are estimated via maximum likelihood estimation by testing different values through multiple iterations to optimize for the best fit of log odds. All of these iterations produce the log likelihood function, and logistic regression seeks to maximize this function to find the best parameter estimates. Given the optimal parameters, the conditional probabilities for each observation can be calculated, logged, and summed together to yield a predicted probability.
from IPython.display import display, HTML
display(HTML("<style>.rendered_html { font-size: 15px; font-family: 'Trebuchet MS'; }</style>"))